The University of Adelaide
School of Economics
Research Paper No. 2009-15
October 2009
Imperfect Information and the Business Cycle
Fabrice Collard, Harris Dellas and Frank Smets
The University of Adelaide, School of Economics Working Paper Series No 0074 (2009–15)
Imperfect information and the business cycle
∗
Fabrice Collard† Harris Dellas‡ Frank Smets§
Abstract
Imperfect information has played a prominent role in modern business cycle theory. We
assess its importance by estimating the New Keynesian (NK) model under alternative informational assumptions. One version focuses on confusion between temporary and persistent
disturbances. Another, on unobserved variation in the inflation target of the central bank. A
third on persistent mis–perceptions of the state of the economy (measurement error). And a
fourth assumes perfect information (the standard NK–DSGE version). We find that imperfect information contains considerable explanatory power for business fluctuations. Signal
extraction seems to provide a conceptually satisfactory, empirically plausible and quantitatively important business cycle mechanism.
JEL class: E32 E52
Keywords: New Keynesian model, imperfect information, signal extraction, Bayesian estimation.
∗
We would like to thank Marty Eichenbaum, Jesper Lindé, Thomas Lubik, Frank Schorfheide and Raf Wouters
for valuable suggestions.
†
School of Economics, The University of Adelaide, SA 5005 Australia. Tel: (+61) (0)8-8303-4928 Fax: (+61)
(0)8- 8223 1460 URL: email: [email protected], Homepage:http://fabcol.free.fr
‡
Department of Economics, University of Bern, CEPR. Address: VWI, Schanzeneckstrasse 1, CH 3012 Bern,
Switzerland. Tel: (41) 31–631–3989, Fax: (41) 31–631–3992, email: [email protected], Homepage:
http://www.vwi.unibe.ch/amakro/dellas.htm
§
Frank Smets: European Central Bank, CEPR and Ghent University, Kaiderstrasse 29 D-60311 Frankfurt am
Main, Germany, Tel: (49)–69–1344 8782 Fax: (49)–69–1344 8553 e-mail: [email protected]
1
Introduction
The role of imperfect information in business fluctuations has been a prominent theme in modern business cycle theory. In the monetary, rational expectations model of Lucas, 1972, misperceptions of nominal aggregates give rise to confusion between nominal and relative price
movements and constitute the main source of economic fluctuations.1 Confusion about the
shocks afflicting the economy is also present in the original version of the RBC model (Kydland
and Prescott, 1982). In this model, the confusion arises from the agents’ inability to distinguish
between temporary and permanent changes in multi-factor productivity.
In spite of its presence in two of the most influential modern macroeconomic models, imperfect
information is not considered an important element for understanding business cycles. There is a
presumption that imperfect knowledge of the true values of nominal aggregates is not important
enough to generate large and persistent movements in real economic activity. And the success
of the Kydland-Prescott RBC model in accounting for business cycle owes little, if anything, to
informational problems.
Recent work, however, has questioned the unimportance of imperfect information/signal extraction problems in influencing macroeconomic outcomes. For instance, Orphanides (2002, 2003)
has forcefully argued that mis–perceptions of potential output may have played the key role behind the excessively loose monetary policy and the great inflation of the 70s. Collard and Dellas,
2007, claim that the failure of the rational expectations literature to find significant effects of
mis-perceived money may be due to its use of inappropriate measures of mis–perceptions. In
particular, using the proper measure leads to the uncovering of significant effects on economic
activity. Moreover, they demonstrate that under –empirically plausible levels of– imperfect information the NK model can deliver considerable inertia. Collard and Dellas, 2006, argue that
the success of imperfect information in accounting for business cycles is not limited to models
with sticky prices. In particular, confusion regarding the types of shocks afflicting the economy
plays a crucial role in allowing also the RBC model to fit a number of conditional, dynamic,
stylized facts. Finally, Woodford, 2003, and Lorenzoni, 2006, show that a Phelps-Lucas island
type of economy with strategic complementarities, idiosyncratic shocks and noisy aggregate
money supply (in the former) or productivity (in the latter), can exhibit realistic business cycle
patterns if the amount of noise is sufficiently high. In a similar vein, Angeletos and La’o, 2009,
argue that incomplete information can be useful for understanding business-cycles.
Why should one expect that imperfect information about the shocks and the true state of the
economy could play an important role in business cycles? The reason is to be found in the
response of the agents to perceived shocks. First, the agents may over–react to some shocks
1
Such mis-perceptions may arise either from the complete unavailability of up-to-date information. Or, from
the fact that the available information is contaminated by measurement error (King, 1982)
2
and this generates “excessive” volatility.2 And second, they under-react to some other shocks.
In particular, the reaction to a particular shock may be initially mooted as the agents attribute
part of the shock to other disturbances. And it may pick up gradually as the agents learn and
update their perceptions of the actual shocks that haver occurred. Imperfect information may
thus provide an endogenous propagation mechanism that can generate inertia, persistence and
reversals, and it can often lead to hump shaped dynamics (see, for instance, Dellas, 2006).
The objective of this paper is to undertake a comprehensive study of the role of imperfect
information in macroeconomic fluctuations. Comprehensive in the sense that alternative informational setups are considered. And that the alternative versions of imperfect information are
compared not only among themselves but also to versions that assume perfect information.
Of the latter class of models we consider the two most widely used ones. Namely, the purely
forward looking version of the NK model (Woodford, 2002). And its popular “hybrid” version
with both forward and backward looking elements as well as real rigidities (Christiano et al,
2005, Galı́ and Gertler, 1999, Smets and Wouters, 2003). Of the class of models with imperfect
information we consider three versions, each one relying on common, imperfect information but
emphasizing a different source of mis–perception. The first one is motivated by Kydland and
Prescott, 1982, and Orphanides, 2002, and involves confusion between temporary and permanent
shocks. The second is motivated by Cogley and Sbordone, 2006, and relies on –unobserved–
variation in the inflation target of the central bank. These two versions share in common the
assumption that the agents observe perfectly all endogenous variables (inflation, output, interest
rate) but not all of the shocks. The third and final version draws on Collard and Dellas’, 2007,
argument that, as revealed by the process of data revisions, very few aggregate variables are
observed accurately. Consequently, unlike the two versions presented above, it assumes that the
shocks as well as some of the endogenous variables are observed with noise.3 This version thus
represents a more severe case of information imperfection than the other two. Nevertheless, as
we will establish later, the degree of mis–perception required by the model in order to fit the
data is well within the range observed in the real world.
The models are estimated on US quarterly data over the 1966–2002 period using Bayesian
methods. They are then evaluated and compared in terms of various criteria: Overall fit (the
log-likelihood), unconditional second moments, and IRFs.4
2
For instance, they respond to changes in monetary aggregates while such changes would have been otherwise
neutral.
3
While our story emphasizes measurement error in aggregate variables, rational inattention à la Sims, 2003,
might also generate similar mis–perceptions. The implications of the latter have not been investigated econometrically yet.
4
These comparisons constitute one of the main differences between this paper and other work in the literature
that tests the empirical validity of the perfect information NK model under alternative specifications. For instance,
Eichenbaum and Fisher, 2004, find that an estimated version of the NK model with backward indexation is
consistent with the data (as judged by the J–statistic in the context of GMM estimation). De Walque, Smets
and Wouters, 2004, find that the Smets and Wouters model performs well even when the parameter of backward
3
The main finding is that imperfect information has considerable explanatory power for the US
business cycle. In particular, it improves the fit of the baseline NK model (that is, the forward
looking model without real rigidities) according to the log-likelihood criterion. Moreover, a
model that exhibits a more “severe” informational problem (the Collard and Dellas version) fares
better with regard to not only the NK model but also the ”hybrid” model (the one with perfect
information, backward indexation and real rigidities) in spite of the fact that it does not itself
contain any real rigidities. This is an important result as backward indexation and real rigidities
have been the subjects of controversy and there is ongoing search in the profession to develop
models that share its success but rely less –at least quantitatively – on these features. Moreover,
we find that the success of the imperfect information model does not require implausible amounts
of noise in the signal extraction problem. Namely, the amount of estimated noise corresponds
closely to the measurement errors observed in real time data.
The remaining of the paper is organized as follows. Sections 1 and 2 present the model with and
without a signal extraction problem (what we call perfect and imperfect information, respectively) and discuss the econometric methodology. Section 3 discusses the econometric methodology. Section 4 presents the main results. The last section offers some concluding remarks.
1
1.1
The model without signal extraction
The Household
There exists an infinite number of households distributed over the unit interval and indexed by
j ∈ [0, 1]. The preferences of household j are given by5
∞
2 X
τ
h ht+τ
Et
β χt+τ log(ct+τ − gt+τ ) − ν
2
(1)
τ =0
where 0 < β < 1 is a constant discount factor, ct denotes consumption in period t, and hjt is the
quantity of labor supplied by the representative household of type j. χt is a preference shock
that is assumed to follow an AR(1) process of the form
log(χt ) = ρχ log(χt−1 ) + (1 − ρχ ) log(χ) + εχ,t
where |ρχ | < 1 and εχ,t
N (0, σχ2 )).
gt denotes an external habit stock which is assumed to be proportional to past aggregate consumption:
gt = ϑct−1 with ϑ ∈ [0, 1).
indexation is close to zero. But the fact that the model without indexation is not rejected by the data does not
mean that it performs the best within a particular class of models.
1+σh
P
(ct+τ −gt+τ )1−σc
5
τ
h ht+τ
β
χ
A more general utility function would be Et ∞
−
ν
. However, it turns
t+τ
τ =0
1−σc
1+σh
out that both σc and σh are not identified in this model. Our specification amounts to setting σc = σh = 1.
4
In each period, household j faces the budget constraint
Bt + Pt ct = Rt−1 Bt−1 + Wt ht + Πt
(2)
where Bt is nominal bonds. Pt , the nominal price of goods. ct denotes consumption expenditures.
Wt is the nominal wage. Ωt is a nominal lump-sum transfer received from the monetary authority
and Πt denotes the profits distributed to the household by the firms.
This yields the following set of first order conditions
νh ht =
χt
ct − θct−1
1.2
1.2.1
wt
ct − θct−1
= βRt Et
(3)
χt+1
(ct+1 − θct )πt+1
(4)
The firms
Final Good Producers
The final good, y is produced by combining intermediate goods, yi , by perfectly competitive
firms. We follow Kimball, 1995, in assuming a production function of the type
Z 1 yt (i)
G
di = 1
yt
0
(5)
where G(·) is an increasing and concave function. Profit maximization gives rise to the following
demand function for good i
yt (i) = yt G
where κt ≡
R1
0
G0
yt (j)
yt
0 −1
Pt (i)
κt
Pt
(6)
yt (j)
yt dj.
The general price index
Z
1
Pt =
G
0 −1
0
1.2.2
Pt (i)
κt di
Pt
(7)
Intermediate goods producers
Each firm i, i ∈ (0, 1), produces an intermediate good by means of labor according to production
function
yt (i) = at ht (i) − Φ
(8)
where ht (i) denotes the labor input used by firm i in the production process. Φ > 0 is a fixed
cost. at is an exogenous technology shock which is assumed to follow an AR(1) process of the
form
log(at ) = ρa log(at−1 ) + (1 − ρa ) log(a) + εa,t
5
where |ρa | < 1 and εa,t
N (0, σa2 ).
Intermediate goods producers are monopolistically competitive, and therefore set prices for the
good they produce. We follow Calvo in assuming that firms set their prices for a stochastic
number of periods. In each and every period, a firm either gets the chance to adjust its price
(an event occurring with probability ξ) or it does not. If it does not get the chance, then it is
assumed, following Christiano et al.,2005, to set prices according to
Pt (i) = πt−1 Pt−1 (i).
(9)
Hence there is perfect indexation to past inflation. In the sensitivity analysis we consider alternative specifications involving partial indexation to past inflation combined with partial indexation
to either the –variable– inflation target of the central bank or steady state inflation.
If a firm i sets its price optimally in period t then it chooses a price, Pt? , in order to maximize:
max Et
Pt? (i)
∞
X
Φt+τ (1 − ξ)τ (Pt? (i)Ξt,τ − Pt+τ Ψt+τ ) yt+τ (i)
τ =0
subject to the total demand (6) and
πt × . . . × πt+τ −1
Ξt,τ =
1
if τ > 1
otherwise
Ψt denotes the real marginal cost of the firm.6 Φt+τ is an appropriate discount factor derived
from the household’s evaluation of future relative to current consumption. This leads to the
first order condition
Et
∞
X
τ =0
1 G0 (ςt+τ )
=0
Φt+τ (1 − ξ)τ yt+τ (i) Pt? (i)Ξt,τ + (Pt? (i)Ξt,τ − Pt+τ Ψt+τ )
ςt+τ G00 (ςt+τ )
where ςt+τ = G0−1
Pt? (i)Ξt,τ
Pt+τ κt+τ
(10)
. Since the price setting scheme is independent of any firm
specific characteristic, all firms that reset their prices will choose the same price, such that
Pt? (i) = Pt? .
In each period, a fraction ξ of contracts ends and (1 − ξ) survives. Hence, from (7) and the price
mechanism, the aggregate intermediate price index writes
? πt−1 Pt−1
0 −1
? 0 −1 Pt
Pt = ξPt G
κt + ξπt−1 G
κt
Pt
Pt
6
(11)
Since there is only one common technology shock and labor is homogenous, the real marginal cost is the same
across firms.
6
1.3
Monetary Policy
Monetary policy is conducted according to
log(Rt ) = ρr log(Rt−1 )+(1−ρr )[log(R? )+log(π t )+αy (log(yt )−log(ytn ))+απ (log(πt )−log(π t )]+r,t
where π t represents the inflation target of the central bank and ytn is the natural rate of output,
that is the output level that would prevail in a flexible price economy. r,t is a monetary policy
shock. It follows a Gaussian iid process (εr,t
N (0, σr2 )).
We will consider two alternative cases for the inflation target. In the first one, the central bank
targets the constant steady state inflation, hence π t = π. In the second one, following Cogley
and Sbordone, 2006, we assume that the inflation target varies over time. In particular, it follows
an AR(1) process of the form
log(π t ) = ρπ log(π t−1 ) + (1 − ρπ ) log(π) + επ,t
where |ρπ | < 1 and επ,t
1.4
N (0, σπ2 ).
Equilibrium
In equilibrium, we have yt = ct . Log-linearization of the model around the deterministic steady
state leads to the standard IS-AS-MP representation of the NK model.
ϑ
1
1−ϑ b
ybt−1 +
Et ybt+1 −
(Rt − Et π
bt+1 ) + x
bt
1+ϑ
1+ϑ
1+ϑ
γ
β
ξ(1 − β(1 − ξ))
2+ϕ−ϑ
π
bt =
π
bt−1 +
Et π
bt+1 +
ybt
1 + βγ
1 + βγ
(1 − ξ)(1 + βγ)(ϕζ + 1) 1 − ϑ
ϑ
ξ(1 − β(1 − ξ))
−
ybt−1 − zbt + νbt
(1 − ξ)(1 + βγ)(ϕζ + 1) 1 − ϑ
bt =ρr R
bt−1 + (1 − ρr )(αy (b
bt ) + r,t
R
yt − ybtn ) + απ π
bt + (1 − απ )π
(1 + ϕ)(1 − ϑ)(1 − ξ)(1 + βγ)(ϕζ + 1)
ϑ(1 + ϕ) n
ybtn =
yb +
zbt
2 + ϕ − ϑ t−1
(2 + ϕ − ϑ)ξ(1 − β(1 − ξ))
ybt =
(12)
(13)
(14)
(15)
where ϕ is the share of fixed costs in total output, and ζ is the curvature of the Kimball goods
market aggregator. x
bt =
1−ϑ
χt
1+ϑ (b
− Et χ
bt+1 ), zbt =
2ξ(1−β(1−ξ))
bt .7
(1−ξ)(1+βγ)(ϕζ+1) a
νbt is a cost push shock
which is assumed to be iid and normally distributed with mean 0 and standard deviation σν .
We will study five versions of the model, two with perfect and three with imperfect information.
In particular, in the first version we assume that: the agents have perfect information about all
shocks; the Central Bank targets a constant inflation rate (σπ = 0, ρπ = 0); there is no indexation
to lagged inflation in price setting (γ = 0); and there are no real rigidities present(ϑ = 0). This
7
Note that both x
bt and zbt follow the same AR(1) process as their underlying fundamentals, χt and at , but the
volatility of the innovations is a simple re–scaling of the original innovations. In order to save on notation, we
will keep σχ and σa to denote the volatilities of the transformed shocks.
7
version, which we will call model 1, corresponds to the standard, baseline New Keynesian (NK)
model.8 The second version differs from the first regarding the price setting scheme as well as
the presence of real rigidities. In particular, here we assume that firms that cannot optimally
reset their prices adopt a backward indexation scheme. And that ϑ is allowed to differ from
zero. This version is denoted as model 2.
The remaining versions introduce imperfect information.9 The third version of the model is
similar to model 1 (γ = σπ = ρπ = ϑ = 0) with the exception that the agents do not observe
the shocks directly and can only infer them through their —perfect— observation of output,
inflation and the nominal interest rate. While the observation of the nominal interest rate
and the level of output enables them to infer the preference and monetary policy shock, the
observation of inflation is not sufficient to help them perfectly distinguish between the persistent
technology and transient cost push shocks. This version, which we call model 3, is —loosely—
motivated by Kydland and Prescott, 1982, and Orphanides, 2002, in the sense that it involves
confusion between temporary and persistent shocks. Under these circumstance, the Central
Bank is assumed to use the best possible estimates of the output gap and the inflation gap it
can produce. This leads us to consider a Taylor rule of the type10
n
b t = ρr R
bt−1 + (1 − ρr )[αy (b
R
yt − ybt|t
) + απ π
bt )] + r,t
(16)
where xt|t denotes the expected value of variable x conditional on the information set of the
Central Bank, which now consists of output, inflation and the nominal interest rate.
The fourth version of the model is similar to the previous one but emphasizes a different source
of mis–perception. The agents still observe the same variables, but the imperfect information
problem does not regard distinguishing between persistent technology shocks and transient cost
push shock, but rather distinguishing between persistent inflation target shocks and transient
monetary policy shocks. This version –which abstracts from the cost push shocks (σν = 0) in
order to be differentiated from version three above– is motivated by the work of Cogley and
Sbordone, 2006, who emphasize the role of –unobserved– variation in the inflation target of the
central bank as a driving force of fluctuations in output and inflation. The central bank is now
8
One could also attempt to estimate the baseline NK model augmented with real rigidities. We run into
identification problems when we tried this. But more importantly, real rigidities are a rather controversial modeling
feature, so it makes sense not to impose them on models that have shunned away from them, such as the baseline
NK model or the imperfect information models. Their exclusion from the latter class of models is particularly
useful as a means of assessing the ability of imperfect information to replace –or ameliorate– all the main inertial
mechanisms of the hybrid NK model.
9
We make the assumption that all agents in the model, including the central bank, have the same information
set. Hence, we abstract from issues of private information and informational asymmetries.
10
Note that since output and inflation are perfectly observed, their perceived value is identical to their actual
value.
8
assumed to follow an interest rate rule of the form:11
n
b t = ρr R
bt−1 + (1 − ρr )[αy (b
R
yt − ybt|t
) + απ (b
πt − π
bt? )] + r,t
(17)
This version is denoted as model 4. Admittedly, the assumption of symmetric information
between the private agents and the monetary authorities is rather tenuous under these circumstances. But the technical complications arising from dropping this assumption are rather
prohibitive in this context.
The last —fifth— version of the model is like the third version, that is, it re–introduces the
cost–push shocks and drops the inflation target shocks. But it adopts a different information
structure that makes the informational problem more severe. In particular, here we assume that
some of the variables of the model (shocks and endogenous variables) are measured imprecisely.
We assume that the shocks, output and the inflation rate are measured with error, that is, for
variable ω we have that
ωt? = ωtT + υω,t
where ωtT denotes the true value of the variable and υω,t is a noisy process that satisfies E(υω,t ) =
0 for all t; E(υω,t εz,t ) = E(υω,t εx,t ) = E(υω,t επ,t ) = E(υω,t εν,t ) = 0; and
2
ηω if t = k
E(υω,t υω,k ) =
0 Otherwise
The nominal interest rate is still observed perfectly. In this version, the Central Bank follows
the rule
n
b t = ρr R
bt−1 + (1 − ρr )[αy (b
R
yt|t − ybt|t
) + απ π
bt|t ] + r,t
(18)
Consequently, the agents face a more severe —and hence, potentially more consequential—
signal extraction problem in the fifth version in comparison to the other imperfect information
versions. The motivation for this specification12 comes from Orphanides, 2002, who argues that
a large fraction of the output gap mis–perception during the great inflation of the 70s can be
attributed to the mis-measurement of actual output. Using the real time data constructed by the
Philadelphia FED, Collard and Dellas, 2007, have argued that such mis–measurement is present
in all macroeconomic series (including GDP inflation) and that it is quantitatively substantial.
The existence of real time data can help assess the plausibility of the estimated amount of noise
in the model by comparing it to that present, say, in data revisions. Or, by comparing perceived
11
Note that since output and inflation are perfectly observed, their perceived values coincide with their actual
values.
12
It should be noted that we are not the first one to estimate a NK model with signal extraction due to
measurement errors. Lippi and Neri, 2006, estimate such a model but, having a different objective, they do not
examine the stochastic properties of their model. Moreover, they do not compare the performance of their model
to that of alternative NK specifications, so one cannot judge its relative success.
9
values to the corresponding real time data.13 Nonetheless, it should be noted that imperfect
observation of the true values of some of the variables of the system does not have to derive
exclusively or even predominantly from measurement error in aggregate variables for the story
to work. An alternative source could be a Sims’, 2003, type of inattention on the part of the
agents.
2
Econometric Methodology
2.1
Data
The model is estimated on US quarterly data for the period 1966:I–2002:IV. The data description
and sources can be found in the appendix. Output is measured by real GDP, inflation is the
annualized quarterly change in the GDP deflator, while the nominal interest rate is the Federal
Funds Rate. The output series is detrended using a linear trend.
Figure 1: US Data
8
6
4
2
%
0
−2
−4
−6
−8
GDP
Inflation
FED Fund Rate
−10
−12
1965
2.2
1970
1975
1980
1985 1990
Quarters
1995
2000
2005
Estimation method
We do not estimate all the parameters of the model as some of them cannot be identified in
the steady state and do not enter the log–linear representation of the economy. This is the case
13
In addition to the measurement error specification of the signal extraction problem presented above there
exists an alternative specification that relies on the distinction between idiosyncratic and aggregate shocks. We
have opted for the former approach because it is much easier to implement and can also be evaluated using
available extraneous information on real time data. The latter approach involves asymmetric information and
higher order expectations and it much harder to solve and test.
10
for the demand elasticity, ζ.14 All the other structural parameters are estimated, including the
steady state level of annualized inflation, π ? , and real interest rate, r? . The discount factor, β,
is then computed using the estimated level of the real interest rate.15 We therefore estimate
the vector of parameters Θ = {ϑ, ξ, r? , π ? , ρr , απ , αy , ρa , ρχ , ρπ , σa , σχ , σr , σν , σπ , ηy , ηπ , ϕ}. Θ is
estimated relying on a Bayesian maximum likelihood procedure. As a first step of the procedure,
the log–linear system (12)–(15) is solved using the Blanchard-Khan method. In the specification
with the signal extraction, the model is solved according to the method outlined in the companion
technical Appendix.16 The one sided Kalman filter is then used on the solution of the model
to form the log–likelihood, Lm ({Yt }Tt=1 , Θ), of each model, m. Once the posterior mode is
obtained by maximizing the likelihood function, we obtain the posterior density function using
the Metropolis–Hastings algorithm (see Lubik and Schorfheide, 2004).
Table 1 presents the –quite diffuse– prior distribution of the parameters.17 The habit persistence
parameter, ϑ, is beta distributed as it is restricted to belong to the (0,1) interval. The average
of the distribution is set to 0.50, which is in line with the prior distribution used by Smets and
Wouters, 2004. The steady state inflation rate and real interest rate have a Γ-distribution with
means 1% and 0.5% per quarter respectively.
The parameter pertaining to the nominal rigidity is distributed according to a beta distribution
as it belongs in the (0,1) interval. The average probability of price resetting, ξ, is set to 0.50,
implying that a firm expects to reset prices on average every two quarters. The confidence
interval, though, encompasses average price resetting from 1.1 to 10 quarters.
The persistence parameter of the Taylor rule, ρr , has a beta prior over (0,1) so as to guarantee
the stationarity of the rule. The prior distribution is centered on 0.5. The reaction to inflation,
απ , and output, αy , is assumed to be positive, and with a normal distribution centered at 1.5
and 0.125 respectively. These values correspond to values commonly used in the literature.
We have little knowledge of the processes that describe the forcing variables. We assume a
beta distribution for the persistence parameters in order to guarantee the stationarity of the
processes. Each distribution is centered on 0.5. Volatility is assumed to follow an inverse
gamma distribution (to guarantee positiveness). However, in order to take into account the
limited knowledge we have regarding these process we impose non informative priors. The same
strategy is also applied to the process of noise in the signal extraction model.
14
Following Smets and Wouters, 2007, we set ζ = 10.
More precisely, we have β = (1 + r? /100)−1/4 .
16
The technical appendix is available from http://fabcol.free.fr/index.php?page=research.
17
In the sensitivity analysis we report results with even more diffuse —essentially uninformative— priors as
a means of establishing that the tightness of the priors used does not matter for the obtained ranking of the
alternative models.
15
11
Table 1: Priors
Param.
θ
ϕ
ξ
r?
π?
ρr
απ
αy
ρa
ρχ
ρπ
σa
σχ
σr
σπ
σν
ηy
ηπ
Type
Beta
Beta
Beta
Gamma
Gamma
Beta
Normal
Normal
Beta
Beta
Beta
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma
Param 1
0.50
0.25
0.50
0.50
1.00
0.50
1.50
0.125
0.50
0.50
0.50
0.20
0.20
0.20
0.20
0.20
0.20
0.20
Param 2
0.25
0.125
0.25
0.50
1.00
0.25
0.25
0.05
0.25
0.25
0.25
4.00
4.00
4.00
4.00
4.00
4.00
4.00
95% HPDI
[0.096,0.903]
[0.067,0.457]
[0.096,0.903]
[0.026,1.501]
[0.051,2.996]
[0.096,0.903]
[1.088,1.909]
[0.045,0.207]
[0.096,0.903]
[0.096,0.903]
[0.096,0.903]
[0.17,0.83]
[0.17,0.83]
[0.17,0.83]
[0.17,0.83]
[0.17,0.83]
[0.17,0.83]
[0.17,0.83]
Note: The parameters are distributed independently from each other. a 95percent highest probability density (HPD) credible intervals. The Param 1 and
Param 2 report the mean and the standard deviation of the prior distribution for
Beta, Gamma and Normal distributions. They report the s and ν parameters
of the inverse gamma distribution, where f (σ|s, ν) ∝ σ −(1+ν) exp(−νs2 /2σ 2 ).
12
3
Estimation Results
Table 2 reports the posterior estimates for the five model versions.18 Figures 2–6 record IRFs
and Table 8 provides information on unconditional moments. The main findings are summarized
below.
First, the estimated parameters are within the range typically found in the literature. The
main exception to this regards the ”persistence” parameters in the hybrid model. In particular,
the persistence parameters of the technology and preference shocks are very low relative to the
values typically reported in the literature (0.04 and 0.26, respectively). At the same time, habit
persistence and the average duration of prices are too high (0.94 and 12 quarters respectively).
Inspection of the model indicates how difficult it is to solve this weak identification problem.19
Moreover, there is information in the sample that helps identify the parameters of the models
as the comparison of the prior to posterior densities reveals. That is, the data are informative.
Second, the models can be ranked in terms of fit as measured by the log–likelihood. Within the
models with perfect information, the hybrid version does better than the baseline NK model.
Hence, the inertial dynamics induced by real rigidities and backward indexation are vital for
empirical success. Within the models of imperfect information, the model with the most severe
signal extraction problem (version 5) does best. Comparing across the two classes of models
one can see that the while the hybrid model does better than models 3 and 4 (which in turn
perform better than the baseline version 1), it falls short of model 5. This indicates that while
imperfect information can be a substitute for real rigidities and backward inflation indexation,
it must be present in a sufficiently large amount in order to be a perfect substitute. To see
this point notice that models 4 and –in particular– 3 lack sufficient persistence in the signal
extraction problem. For instance, consider model 3. The complete absence of serial correlation
in the cost–push shock allows the agents to quickly discriminate between this and the persistent
technology shock, leaving little room for inertial dynamics. Inspection of the corresponding
IRFs confirms this. A more direct confirmation can be derived when one compares the paths of
actual and perceived shocks (see Figure 7 in the Appendix). As can be seen, mis–perceptions
are resolved fairly fast. Hence, introducing a limited signal extraction problem to the baseline
NK model and expecting it to outperform the hybrid model may amount to asking for too much.
Nevertheless, making the signal extraction problem somewhat more severe/persistent does the
18
The interested reader may consult the technical appendix that accompanies the paper for more details regarding the distribution of posterior estimates.
19
Note that setting the indexation parameter, γ equal to one ameliorates the identification problem significantly
as it allows the Metropolis–Hastings algorithm to converge. But it does not manage to solve the weak identification
problem. Imposing fixed values for some of the persistence parameters would take care of the problem but might
prove unduly restrictive and also undermine the performance of the hybrid model. For instance, when fixing the
persistence in the technology and preference shocks to 0.95, the parameter of habit persistence decreases –to 0.52–
and the Calvo parameter increases to 0.38. But marginal likelihood also drops to -277.
13
Table 2: Estimation Results
Parameter
θ
(1)
–
(2)
0.94
(3)
–
(4)
–
(5)
–
ξ
0.49
0.08
0.49
0.68
0.22
[0.33,0.64]
[0.03,0.14]
[0.34,0.64]
[0.51,0.83]
[0.14,0.31]
[0.90,0.97]
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
ρπ
0.26
0.29
0.27
0.24
0.28
[0.04,0.48]
[0.07,0.51]
[0.05,0.49]
[0.02,0.46]
[0.06,0.50]
0.61
0.66
0.62
0.56
0.62
[0.31,0.86]
[0.36,0.97]
[0.34,0.87]
[0.11,0.92]
[0.22,1.02]
0.90
1.01
0.91
0.86
0.97
[0.53,1.24]
[0.69,1.33]
[0.57,1.26]
[0.34,1.30]
[0.57,1.34]
0.67
0.84
0.67
0.21
0.27
[0.60,0.75]
[0.77,0.91]
[0.60,0.74]
[0.02,0.39]
[0.14,0.42]
1.61
1.40
1.58
2.37
1.64
[1.29,1.90]
[1.06,1.75]
[1.31,1.90]
[2.05,2.70]
[1.19,2.10]
0.13
0.04
0.15
0.13
0.20
[0.04,0.23]
[0.00,0.12]
[0.05,0.24]
[0.03,0.22]
[0.11,0.28]
0.98
0.07
0.98
0.99
0.95
[0.96,1.00]
[0.00,0.18]
[0.96,1.00]
[0.97,1.00]
[0.92,0.97]
0.91
0.26
0.91
0.92
0.86
[0.85,0.97]
[0.09,0.42]
[0.85,0.97]
[0.85,0.98]
[0.80,0.92]
–
–
–
0.92
–
[0.86,0.98]
σa
σχ
σr
σν
σπ
0.27
0.11
0.25
0.81
0.12
[0.14,0.44]
[0.08,0.14]
[0.12,0.40]
[0.35,1.38]
[0.09,0.16]
0.16
0.62
0.17
0.20
0.29
[0.11,0.22]
[0.48,0.77]
[0.12,0.23]
[0.16,0.24]
[0.21,0.37]
0.36
0.26
0.35
0.49
0.13
[0.30,0.41]
[0.23,0.29]
[0.30,0.41]
[0.37,0.63]
[0.09,0.16]
0.14
0.15
0.16
–
[0.10,0.19]
[0.11,0.18]
[0.11,0.22]
–
–
–
0.21
[0.18,0.25]
0.11
–
[0.08,0.15]
ηy
–
–
–
–
0.28
[0.10,0.52]
ηπ
–
–
–
–
L
-323.472
-267.388
-321.101
-296.148
6.72
[2.18,13.15]
-264.729
Note: This table reports the mean of the posterior distribution of each parameter and
the associated 95% HPDI (in brackets). (1): Baseline NK, (2): Hybrid NK (Backward
Indexation, Real Rigidities), (3): Imperfect Info. Temporary vs Permanent Shocks, (4):
Imperfect Info., Cogley–Sbordone, (5): Imperfect Info., Noisy Signals. L denotes the average log marginal density of the model. 95% HPDI in brackets.
14
Table 3: Moments (HP–filtered series)
σy
Data
1.56
σπ
0.30
σR
0.44
ρ(π, y)
0.12
ρ(R, y)
0.34
ρy (1)
0.87
ρπ (1)
ρR (1)
0.48
0.82
ρy (2)
0.69
ρπ (2)
0.31
ρR (2)
0.58
ρy (4)
0.27
ρπ (4)
0.24
ρR (4)
0.25
(1)
1.47
(2)
1.56
(3)
1.45
(4)
1.21
(5)
1.15
[1.27,1.67]
[1.26,1.89]
[1.25,1.64]
[1.07,1.36]
[1.02,1.28]
0.38
0.45
0.37
0.35
0.30
[0.33,0.43]
[0.40,0.51]
[0.33,0.42]
[0.31,0.39]
[0.26,0.33]
0.37
0.49
0.37
0.43
0.35
[0.33,0.41]
[0.40,0.59]
[0.33,0.41]
[0.38,0.48]
[0.31,0.40]
0.24
0.19
0.26
0.09
0.21
[0.12,0.36]
[0.05,0.34]
[0.13,0.38]
[0.02,0.16]
[0.13,0.29]
-0.25
-0.21
-0.25
-0.05
0.26
[-0.37,-0.13]
[-0.40,-0.01]
[-0.36,-0.13]
[-0.12,0.02]
[0.11,0.41]
0.65
0.85
0.63
0.71
0.68
[0.63,0.68]
[0.81,0.89]
[0.60,0.66]
[0.69,0.72]
[0.66,0.71]
0.37
0.72
0.40
0.42
0.33
[0.30,0.45]
[0.69,0.75]
[0.33,0.46]
[0.33,0.52]
[0.22,0.43]
0.65
0.81
0.65
0.73
0.71
[0.59,0.70]
[0.77,0.86]
[0.60,0.71]
[0.69,0.76]
[0.66,0.77]
0.41
0.64
0.39
0.47
0.43
[0.38,0.44]
[0.58,0.71]
[0.34,0.42]
[0.46,0.48]
[0.40,0.47]
0.13
0.47
0.14
0.25
0.20
[0.07,0.20]
[0.42,0.50]
[0.08,0.21]
[0.18,0.31]
[0.12,0.27]
0.40
0.61
0.40
0.47
0.44
[0.33,0.46]
[0.55,0.67]
[0.34,0.46]
[0.43,0.51]
[0.38,0.51]
0.08
0.25
0.07
0.12
0.08
[0.06,0.10]
[0.16,0.34]
[0.04,0.10]
[0.11,0.12]
[0.05,0.11]
-0.06
0.08
-0.06
0.03
0.01
[-0.08,-0.03]
[0.03,0.13]
[-0.09,-0.03]
[-0.00,0.05]
[-0.01,0.04]
0.06
0.22
0.06
0.09
0.06
[0.01,0.10]
[0.16,0.29]
[0.01,0.10]
[0.05,0.13]
[0.01,0.11]
Note: (1): Baseline NK, (2): Hybrid NK (Backward Indexation, Real Rigidities), (3):
Imperfect Info. Temporary vs Permanent Shocks, (4): Imperfect Info., Cogley–Sbordone,
(5): Imperfect Info., Noisy Signals. 95% HPDI in brackets.
15
job. Namely, it gives the baseline model the feature it needs in order to match the data well.
The comparisons of the IRFs and moments across models offers some clues regarding not only
the reasons for the differences in their relative performance but also about the driving forces in
each model. The most successful models (2,4 and 5) tend to exhibit inertia in inflation dynamics
while the less successful ones (1 and 3) do not. Good performance at this front seems to play
an important role, as the differences in performance across models in the realm of unconditional
moments tend to be quite small (see Table 3, which follows standard practice and used HP
filtered data, or Table 8 in the appendix which used linearly detrended data. ).
Note that the main contributor to inflation inertia differs across the three specifications. In
the perfect information model with indexation, inflation inertia comes from the two demand
disturbances (the preference and the monetary policy shock). In the model with the variable
inflation target, it arises exclusively from the inflation target shock while in the model with
measurement errors, from the preference shock. Hence the three models differ regarding their
identification of the cause of inertia between —direct— policy and non–policy factors.
The mechanism for inflation inertia in the model of indexation is well known, so let us examine
the mechanisms in the models with imperfect information. Figures 9–10 plot the actual and
perceived paths of the shocks in the fourth and fifth models. As can be seen, the agents under–
react to the shocks. It also takes them considerable time to recognize the true nature of the shock
that has afflicted the economy. For instance, consider a positive inflation target shock (a higher
inflation target). If the shock were perfectly observable, everybody’s expectation and hence
actual inflation would jump up and then, given the AR(1) assumption on the shock, it would
monotonically decline. But if the shock is not observed then the agents cannot tell whether the
lower nominal interest rate and higher inflation should be expected to persist in the future or
not (as it would not if the shock represented a purely transient monetary disturbance (εe,t )).
As time goes one and the agents continue observing a lower nominal interest rate they become
more convinced that an inflation target shock has taken place and inflation increase relative to
the initial period. Due to the AR(1) nature of the shock, it reaches a peak after a few periods
and then starts declining towards the initial steady state.
Judging the mis–perceptions story
Models with measurement errors and signal extraction are often criticized as requiring implausibly large amounts of imperfect information in order to deliver good results. In model 5 we
have attributed the mis–perception problem to measurement errors associated with preliminary
data releases. In order to judge the contribution of this mechanism, one could compare the
properties of the noise in the model to that present in real time data. A natural way of doing so
is by computing the volatility of the data revisions in the real world (for instance, initial minus
final release) and the volatility between the actual and the perceived values in the model. The
16
underlying assumption here is that the actual value corresponds to the final revision while the
perceived value to the preliminary release. Hence, the revision in, say, GDPt is, log(Yt|t )−log(Yt )
where the former represents the initial and the latter the final release.
Table 4 presents information on the properties of these revisions.20
Table 4: The properties –moments– of data revisions
σεπ
Whole Sample
Data
Model
0.22,1.51
0.94
σε∆y
0.54,0.68
ρ(επ , π)
0.14,-0.09
ρ(ε∆y , ∆y)
0.38,0.28
Post 1982
Data
Model
0.19,1.97
0.78
[0.83,1.05]
0.51
[0.66,0.91]
0.40,0.58
[0.41,0.61]
0.24
[0.29,0.46]
0.08,-0.12
[0.14,0.33]
0.86
0.12,0.00
-0.03
-0.09,-0.12
-0.03
[-0.04,-0.02]
0.74
[0.60,0.88]
0.04,-0.02
[-0.04,-0.02]
ρε∆y (1)
0.23
[0.15,0.32]
0.53,0.28
[0.78,0.93]
ρεπ (1)
0.37
-0.04
[-0.06,-0.03]
-0.16,-0.24
-0.03
[-0.04,-0.02]
Note: επ is the revision in inflation and ε∆y in output growth. σ is standard
deviation, ρ is cross-correlation and ρ(1) is 1st order autocorrelation. The first
number in column 1 and 3 corresponds to the computation from a single vintage,
and the second number to the computation from two successive vintages (for
details see the footnote). 95% HPDI in brackets.
Several observations are in order. First, the amount of noise the model needs in fitting the
data is not excessive relative to that present in data revisions. Second, the assumption made in
the model that the correlation between the measurement error and the initial release is zero is
satisfied in the data for inflation but not for output. In other words, the initial release seems to
contain information about future releases, or equivalently, the measurement error is correlated
with the business cycle. This is an interesting observation that has implications for the ability
of the “noisy aggregate variables” theory of the business cycle to fit the data. In particular,
it seems quite conceivable that incorporating a more realistic model of measurement error in
model 5 could further improve its performance. While pursuing this track is outside the scope
of the present paper it certainly represents a promising line of future research. And third, there
20
Some caveats are in order. First, there is a difference in timing between the real world and the model. In
particular, in the model we assume that the observation xt|t is available in period t. In the real world, data are
released with a one period lag, that is, xt|t+1 rather than xt|t is the data available. And second, in computing
growth rates one may either use data from the same vintage (which then already includes a revision) or from two
successive vintages. That is, either ∆xt|t = log(xt|t ) − log(xt−1|t ) or ∆xt|t = log(xt|t ) − log(xt−1|t−1 ). It is not
obvious which measure is more relevant for testing mis–perceptions in a model where agents’ decisions involve
levels of variables and there is no lag in the release of information. Consequently, we have chosen to report both
measures.
17
is no serial correlation in the revisions.
4
Sensitivity analysis
In this section we discuss the implications of alternative specifications in order to judge the
robustness of the main results and rankings reported above. The likelihood results from these
exercises are reported in Tables 5 and 6.21
A natural question regards the importance of the distribution of the priors for the results obtained. It is should be clear from inspection of Table 1 that the priors we employ in the
estimation above are quite diffuse. For instance, the 95% confidence interval for the Calvo
parameter encompasses average price resetting from 1.1 to 10 quarters. Nonetheless, it may
be worth repeating the analysis with even –essentially uninformative– flatter priors. Table 7
reports the new priors. Table 5 reports the posterior likelihood attained by each model both in
the benchmark case and when diffuse priors are used. Inspection of the table shows an increase
in the log–likelihood for all models, which indicates that the choice of the degree of tightness of
the priors does matter. Nonetheless, the ranking of the alternative models is identical to that
reported before. This is encouraging and should help dispel any concerns that our results may
have been driven by the choice of the distribution of the priors.
Table 5: Likelihood: Sensitivity Analysis (Tightness of Priors, Period of Estimation)
Model
(1)
(2)
(3)
(4)
(5)
Benchmark
-323.472
-267.388
-321.101
-296.148
-264.729
Diffuse Priors
-313.883
-258.469
-317.726
-276.183
-255.363
Post–1982 period
-111.093
-111.131
-108.155
-109.042
-107.385
Monetary policy in the models considered here is perfectly credible. Consequently, and to the
extent that the monetary authorities have a known, long term, inflation target, it is sensible to
assume that the non-optimizing firms in model 2 would select to index their prices not only to
past inflation but also to the inflation target itself. We have thus repeated the analysis using
γ
the price setting scheme Pt (i) = πt−1
π 1−γ
Pt−1 (i), where π t , the inflation target of the central
t
bank, follows an AR(1) process of the form:
log(π t ) = ρπ log(π t−1 ) + (1 − ρπ ) log(π) + επ,t .
The second row in Table 6 presents the outcome of the estimation of this variant of model 2.
21
More detailed results are reported in a companion technical appendix which is available on the authors’ web
pages.
18
There is no improvement relative the benchmark case reported above.
Allowing for partial indexation to past inflation also allows us to address a possible concern one
might have from our having imposed perfect indexation (γ = 1) in the estimation in section
3. If the backward price indexation component is less than unity, then imposing a value of
unity could disadvantage the hybrid model vis a vis the other models. We have therefore
carried out the exercise for alternative values of the backward price indexation parameter22
(γ = 0.25, 0.50, 0.75). Again there is no improvement in the performance of the model.
Table 6: Likelihood: Sensitivity Analysis (Hybrid NK Model)
Model
Benchmark
Past Inflation (γ=0.25)
Past Inflation (γ=0.50)
Past Inflation (γ=0.75)
C.B. Inflation Target (γ=0.58)
Likelihood
-267.388
-281.865
-279.587
-275.926
-278.186
For model 5, we have assumed that the agents observe both output and inflation with error.
With only three endogenous variables this is necessary in order to make the signal extraction
problem severe enough and give imperfect information a fighting chance to play a role in shaping
the properties of the model. Nonetheless, it is of interest to see how the properties of the model
would be affected if we minimized the role of mis–perceptions by allowing agents to also perfectly
observe the inflation rate alongside the nominal interest rate.23 In this case, the performance
of this version of the model deteriorates significantly as the signal extraction problem becomes
quite trivial. Our view is that the important issue here is not actually whether the inflation
rate is accurately, contemporaneously observed or not but rather whether there is enough noise
present in the system to make signal extraction a difficult task. A model with more variables
could possess this property even if inflation were made observed with little noise as long as other
endogenous variables were noisy and the structure of the model were such that the observation
of inflation did not undermine the signal extraction problem.
A final sensitivity check carried out was to estimate the models for the post 1982 period. As
is well known, this period has been associated with a significant decline in the volatility of
inflation and output and, as revealed by IRFs of inflation, the reduction of inertia in inflation.
This implies that a model, such as the hybrid, whose additional features relative to the baseline
NK model aim to better capture inertia and volatility may not perform as well as in the full
sample period. The last column of Table 5 shows that this conjecture is correct. The hybrid
model does not improve upon the baseline model during that period. Table 5 also shows that the
22
23
γ
Price indexation now takes the form: Pt (i) = πt−1
π 1−γ Pt−1 (i), where π denotes steady state inflation.
More detailed results are reported in the accompanying appendix.
19
imperfect information models do better than the two perfect information models. It is tempting
to claim that these findings suggest that imperfect information may represent a more structural
business cycle mechanism than other, commonly employed features (such as real rigidities or
backward indexation). Nonetheless, the differences in the marginal likelihood across models are
rather small, so this claim remains tenuous and deserves further investigation.
5
Conclusions
The concepts of imperfect information and mis–perceptions have a distinguished presence in
macroeconomic theory going back to Lucas’, 1972, and Kydland and Prescott’s 1982, seminal
work. Notwithstanding this history and also in spite of Orphanides’ influential work on the
role of mis–perceptions in the great inflation of the 70s, macroeconomic theory has more or less
stayed clear of them. In this paper we have undertaken a comprehensive analysis of its role
in US business cycles during the last 40 years using the New Keynesian model as the vehicle
for the analysis. In particular, we have examined the performance of the model under different
structures of information which are motivated by influential suggestions in the literature, such
as unobserved variation in inflation target (Cogley and Sbordone, 2006) or, confusion between
transitory and persistent real shocks (Kydland and Prescott, 1982, Orphanides, 2002). We found
that imperfect information plays an important role in accounting for the US business cycle and it
can help the NK model match key dynamic properties of the data (inertia, persistence) without
any need for other, popular inertial features (such as real rigidities and backward price indexation schemes. Moreover, its success is not confined to a sample that also includes “unusual”
macroeconomic behavior (the 70s) but it also extends to a period that only contains the great
moderation. This makes us speculate that imperfect information may represent a structural
business cycle mechanism than applies to a variety of macroeconomic environments.
Importantly, this good performance does not hinge on implausibly large amount of informational
frictions. Even the version with the most severe information problems (the one with measurement errors) requires a degree of mis–perception that is well within the range observed in real
time data. Nonetheless, given the fact that measurement errors in the real world do not fully
conform to the standard assumption of independence made in the model, a fruitful line of future
research may involve building more involved models of measurement error and incorporating
them into the model in order to judge whether such errors truly constitute an important source
of mis–perceptions.
20
6
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21
Sims, Christopher, 2003, “Implications of rational inattention,” Journal of Monetary Economics,
50, pp. 665690.
Smets, Frank and Raf Wouters, 2003, “An Estimated Stochastic Dynamic General Equilibrium
Model of the Euro Area”, Journal of European Economic Association, 1,pp. 1123–1175.
Smets, Frank and Raf Wouters, 2007, “Shocks and Frictions in US Business Cycles: A Bayesian
DSGE Approach,” American Economic Review, 97(3), pp. 586–606.
Woodford, Michael, 2003, “Imperfect Common Knowledge and the Effects of Monetary Policy,”
in: P. Aghion, R. Frydman, J. Stiglitz and M. Woodford, Eds, Knowledge, Information, and
Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps, Princeton University
Press, Princeton, pp. 2558.
Woodford, Michael, 2003, Interest and prices, Princeton University Press.
22
A
Data
output = log(GDPC96/LNSindex )×100
inflation = log(GDPDEF/GDPDEF(-1))×100
interest rate = Federal Funds Rate/4
Source of the original data:
• GDPC96 : Real Gross Domestic Product - Billions of Chained 1996 Dollars, Seasonally
Adjusted Annual Rate Source: U.S. Department of Commerce, Bureau of Economic Analysis.
• GDPDEF : Gross Domestic Product - Implicit Price Deflator - 1996=100, Seasonally
Adjusted Source: U.S. Department of Commerce, Bureau of Economic Analysis
• Federal Funds Rate : Averages of Daily Figures - Percent Source: Board of Governors of
the Federal Reserve System
• LNS10000000 : Labor Force Status : Civilian noninstitutional population - Age : 16 years
and over - Seasonally Adjusted - Number in thousands Source: U.S. Bureau of Labor
Statistics (Before 1976: LFU800000000 : Population level - 16 Years and Older)
• LNSindex : LNS10000000(1992:3)=1
23
B
Figures 1966–2002
24
Figure 2: Impulse Response Functions – Perfect info, forward NK
(a) Technology Shock
% deviation
Output
Inflation Rate
Nominal Interest Rate
1.4
0
0
1.2
−0.02
−0.02
1
−0.04
−0.04
0.8
−0.06
−0.06
0.6
−0.08
−0.08
0.4
0
10
Quarters
20
−0.1
0
10
Quarters
20
−0.1
0
10
Quarters
20
(b) Preference Shock
% deviation
Output
Inflation Rate
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
Nominal Interest Rate
0.25
0.2
0.15
0.1
0
0
10
Quarters
20
0.05
0
0
10
Quarters
0
0
20
10
Quarters
20
(c) Interest Rate Shock
% deviation
Output
Inflation Rate
0
0
−0.2
−0.1
−0.4
−0.2
−0.6
−0.3
Nominal Interest Rate
0.25
0.2
0.15
0.1
−0.8
0
10
Quarters
20
−0.4
0
0.05
10
Quarters
20
0
0
10
Quarters
20
(d) Cost Push Shock
% deviation
Output
Inflation Rate
Nominal Interest Rate
0
0.15
0.08
−0.05
0.1
0.06
−0.1
0.05
0.04
−0.15
0
0.02
−0.2
0
10
Quarters
20
−0.05
0
10
Quarters
25
20
0
0
10
Quarters
20
Figure 3: Impulse Response Functions – Perfect info, hybrid NK
(a) Technology Shock
% deviation
Output
Inflation Rate
1
0.2
0.8
0.1
0.6
0
0.4
−0.1
0.2
−0.2
Nominal Interest Rate
0.1
0
−0.1
0
0
10
Quarters
−0.3
0
20
−0.2
10
Quarters
20
−0.3
0
10
Quarters
20
(b) Preference Shock
Output
Inflation Rate
% deviation
1.5
1
Nominal Interest Rate
0.08
0.12
0.06
0.1
0.08
0.04
0.06
0.02
0.5
0.04
0
0
0
10
Quarters
20
−0.02
0
0.02
10
Quarters
0
0
20
10
Quarters
20
(c) Interest Rate Shock
% deviation
Output
Inflation Rate
0.1
0
0
−0.01
Nominal Interest Rate
0.3
0.2
−0.02
−0.1
−0.03
−0.2
−0.04
0
−0.3
−0.4
0
0.1
−0.05
10
Quarters
20
−0.06
0
10
Quarters
20
−0.1
0
10
Quarters
20
(d) Cost Push Shock
Output
Inflation Rate
0
% deviation
−0.2
Nominal Interest Rate
0.4
0.25
0.3
0.2
0.2
0.15
0.1
0.1
0
0.05
−0.4
−0.6
−0.8
0
10
Quarters
20
−0.1
0
10
Quarters
26
20
0
0
10
Quarters
20
Figure 4: Impulse Response Functions – Imperfect info, Pers. vs Temp
(a) Technology Shock
% deviation
Output
Inflation Rate
1.4
0
1.2
−0.02
Nominal Interest Rate
0
−0.02
−0.04
1
−0.06
0.8
−0.04
−0.08
−0.06
0.6
−0.1
0.4
0
10
Quarters
20
−0.12
0
10
Quarters
20
−0.08
0
10
Quarters
20
(b) Preference Shock
% deviation
Output
Inflation Rate
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
Nominal Interest Rate
0.25
0.2
0.15
0.1
0
0
10
Quarters
20
0
0
0.05
10
Quarters
0
0
20
10
Quarters
20
(c) Interest Rate Shock
% deviation
Output
Inflation Rate
0
0
−0.2
−0.1
−0.4
−0.2
−0.6
−0.3
Nominal Interest Rate
0.25
0.2
0.15
0.1
−0.8
0
10
Quarters
20
−0.4
0
0.05
10
Quarters
20
0
0
10
Quarters
20
(d) Cost Push Shock
Output
Inflation Rate
0.2
0.05
0.04
0
% deviation
Nominal Interest Rate
0.15
0.1
0.03
−0.2
0.05
0.02
−0.4
0.01
0
−0.6
−0.8
0
0
10
Quarters
20
−0.05
0
10
Quarters
27
20
−0.01
0
10
Quarters
20
Figure 5: Impulse Response Functions– Imperfect info, Inflation target shock
(a) Technology Shock
Output
Inflation Rate
1.2
% deviation
1
Nominal Interest Rate
0
0
−0.005
−0.01
−0.01
−0.02
−0.015
−0.03
−0.02
−0.04
0.8
0.6
0.4
0
10
Quarters
20
−0.025
0
10
Quarters
20
−0.05
0
10
Quarters
20
(b) Preference Shock
Output
Inflation Rate
0.12
Nominal Interest Rate
0.2
0.4
0.15
0.3
0.1
0.2
0.05
0.1
% deviation
0.1
0.08
0.06
0.04
0.02
0
0
10
Quarters
20
0
0
10
Quarters
0
0
20
10
Quarters
20
(c) Interest Rate Shock
% deviation
Output
Inflation Rate
Nominal Interest Rate
0.1
0
0.05
0
−0.1
0
−0.1
−0.2
−0.05
−0.2
−0.3
−0.1
−0.3
0
10
Quarters
20
−0.4
0
10
Quarters
20
−0.15
0
10
Quarters
20
(d) Inflation Target Shock
Output
Inflation Rate
0.12
Nominal Interest Rate
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
% deviation
0.1
0.08
0.06
0.04
0.02
0
0
10
Quarters
20
0
0
10
Quarters
28
20
−0.05
0
10
Quarters
20
Figure 6: Impulse Response Functions – Imperfect info, measurement errors
(a) Technology Shock
% deviation
Output
Inflation Rate
Nominal Interest Rate
0.2
0
0
0.15
−0.05
−0.005
0.1
−0.1
−0.01
0.05
−0.15
−0.015
0
0
10
Quarters
20
−0.2
0
10
Quarters
20
−0.02
0
10
Quarters
20
(b) Preference Shock
Output
Inflation Rate
0.8
Nominal Interest Rate
0.12
0.25
0.1
0.2
% deviation
0.6
0.08
0.4
0.15
0.06
0.1
0.04
0.2
0.05
0.02
0
0
10
Quarters
20
0
0
10
Quarters
20
0
0
10
Quarters
20
(c) Interest Rate Shock
Output
Inflation Rate
0
0.2
−0.002
−0.05
% deviation
Nominal Interest Rate
0
0.15
−0.004
−0.1
0.1
−0.006
−0.15
−0.2
0
0.05
−0.008
10
Quarters
20
−0.01
0
10
Quarters
20
0
0
10
Quarters
20
(d) Cost Push Shock
Output
0
Inflation Rate
Interest Rate
x Nominal
10
−3
0.3
6
−0.01
5
% deviation
0.2
−0.02
4
−0.03
0.1
3
−0.04
2
0
−0.05
−0.06
0
1
10
Quarters
−0.1
20
0
10
Quarters
29
20
0
0
10
Quarters
20
The degree of mis-perceptions of shocks
Figure 7: IRF: Actual vs Perceived — Imperfect info, Pers. vs Temp
(a) Technology Shock
% deviation
Output Gap
Inflation Rate
0.1
0
0
−0.02
−0.1
−0.04
−0.2
−0.06
−0.3
0
5
10
Quarters
15
20
−0.08
0
5
10
Quarters
15
20
15
20
(b) Cost Push Shock
Output Gap
Inflation Rate
0.2
0.06
0.04
0
% deviation
B.1
0.02
−0.2
0
−0.4
−0.6
0
−0.02
5
10
Quarters
15
20
−0.04
0
5
10
Quarters
Note: Plain line: True, Dashed line: Perceived
30
Figure 8: IRF: Actual vs Perceived — Imperfect info, Pers. vs Temp.
(a) Technology Shock
Techno.
Cost Push
0.24
0
% deviation
% deviation
0.22
0.2
0.18
−0.02
−0.04
−0.06
0.16
0
5
10
Quarters
15
−0.08
0
20
5
10
Quarters
15
20
15
20
(b) Cost Push Shock
Techno.
Cost Push
0.2
0
0.15
% deviation
% deviation
−0.02
−0.04
−0.06
−0.08
0.1
0.05
0
−0.1
−0.12
0
5
10
Quarters
15
−0.05
0
20
5
10
Quarters
Note: Plain line: True, Dashed line: Perceived
Figure 9: IRF: Actual vs Perceived — Imperfect info, Inflation target shock
(a) Monetary Policy Shock
Mon. Pol.
Target
0.6
0
−0.02
% deviation
% deviation
0.4
0.2
0
−0.04
−0.06
−0.08
−0.1
−0.2
0
5
10
Quarters
15
20
−0.12
0
5
10
Quarters
15
20
15
20
(b) Inflation Target Shock
Mon. Pol.
Target
0.12
0
% deviation
% deviation
0.1
−0.05
−0.1
0.08
0.06
0.04
0
5
10
Quarters
15
20
0.02
0
5
10
Quarters
Note: Plain line: True, Dashed line: Perceived
31
Figure 10: IRF: Actual vs Perceived — Imperfect info, Measurement errors
(a) Technology Shock
Techno.
0.12
0
x 10
Pref.
−6
0
0.1
x 10
−4
Cost Push
−0.5
0.08
0.06
0.04
% deviation
% deviation
% deviation
−2
−4
−1
−1.5
−2
−6
0.02
−2.5
0
0
10
Quarters
−8
0
20
10
Quarters
−3
0
20
10
Quarters
20
(b) Preference Shock
Techno.
−6
0
x 10
Pref.
0.4
8
−1
−4
% deviation
% deviation
% deviation
−3
0.2
0.1
10
Quarters
0
0
20
Cost Push
4
2
0
−5
−6
0
−6
6
0.3
−2
x 10
10
Quarters
−2
0
20
10
Quarters
20
(c) Cost Push Shock
−3
0
x 10
Techno.
15
x 10
−6
Pref.
Cost Push
0.3
−1
0.2
% deviation
% deviation
% deviation
10
−0.5
5
0
−1.5
0
10
Quarters
20
−5
0
0.1
0
10
Quarters
20
−0.1
0
Note: Plain line: True, Dashed line: Perceived
32
10
Quarters
20
Figure 11: IRF: Actual vs Perceived — Imperfect info, Measurement errors
(a) Technology Shock
Output Gap
Inflation Rate
0
0
−0.02
% deviation
−1
−0.04
−2
−0.06
−0.08
−3
−0.1
−4
0
5
10
Quarters
15
−0.12
0
20
5
10
Quarters
15
20
15
20
15
20
(b) Preference Shock
Output Gap
Inflation Rate
0.8
0.1
0.08
% deviation
0.6
0.06
0.4
0.04
0.2
0.02
0
0
5
10
Quarters
15
0
0
20
5
10
Quarters
(c) Cost Push Shock
% deviation
Output Gap
Inflation Rate
0.01
0.25
0
0.2
−0.01
0.15
−0.02
0.1
−0.03
0.05
−0.04
0
5
10
Quarters
15
20
0
0
5
10
Quarters
Note: Plain line: True, Dashed line: Perceived
33
Technical Appendix
Not Intended for Publication
A
Robustness Analysis
Table 7: Diffuse Priors
Param.
ϑ
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
ρπ
σa
σχ
σr
σπ
σν
ηy
ηπ
Type
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Normal
Normal
Uniform
Uniform
Uniform
Invgamma
Invgamma
Invgamma
Invgamma
Invgamma
Invgamma
Invgamma
Param 1
0.00
0.00
0.00
0.00
0.00
0.00
1.50
0.125
0.00
0.00
0.50
0.20
0.20
0.20
0.20
0.20
0.20
0.20
Param 2
1.00
1.00
1.00
4.00
4.00
1.00
0.50
0.05
1.00
1.00
0.50
4.00
4.00
4.00
4.00
4.00
4.00
4.00
95% HPDI
[0.025;0.975]
[0.025;0.975]
[0.025;0.975]
[0.10;3.90]
[0.10;3.90]
[0.025;0.975]
[0.52;2.47]
[0.027;0.222]
[0.025;0.975]
[0.025;0.975]
[0.025;0.975]
[0.10;0.38]
[0.10;0.38]
[0.10;0.38]
[0.10;0.38]
[0.10;0.38]
[0.10;0.38]
[0.10;0.38]
Note: The parameters are distributed independently from each other.
a
95-percent highest probability density (HPD) credible intervals
(see ?, p.57). The Param 1 and Param 2 report the lower and
upper bounds for Uniform distributions, the mean and the standard
deviation for the Normal distributions. They report the s and ν
parameters of the inverse gamma distribution, where f (σ|s, ν) ∝
σ −(1+ν) exp(−νs2 /2σ 2 ).
34
Table 8: Moments: Data linearly detrended
σy
Data
3.37
σπ
0.62
σR
ρ(π, y)
0.79
-0.27
ρ(R, y)
-0.39
ρy (1)
0.97
ρπ (1)
ρR (1)
0.87
0.93
ρy (2)
0.92
ρπ (2)
0.83
ρR (2)
0.84
ρy (4)
0.79
ρπ (4)
0.79
ρR (4)
0.69
(1)
5.87
(2)
4.09
(3)
6.11
(4)
6.76
(5)
2.50
[3.11,9.86]
[2.49,5.94]
[2.95,10.39]
[3.13,12.08]
[2.00,3.09]
0.55
0.70
0.54
0.62
0.51
[0.45,0.68]
[0.57,0.85]
[0.44,0.67]
[0.48,0.81]
[0.41,0.61]
0.74
0.89
0.72
0.93
0.60
[0.55,1.00]
[0.70,1.10]
[0.53,0.97]
[0.63,1.33]
[0.48,0.74]
-0.18
0.08
-0.17
-0.06
-0.01
[-0.35,-0.04]
[-0.26,0.37]
[-0.32,-0.03]
[-0.10,-0.02]
[-0.15,0.15]
-0.32
-0.36
-0.31
-0.13
0.03
[-0.52,-0.12]
[-0.65,-0.05]
[-0.50,-0.12]
[-0.23,-0.04]
[-0.17,0.27]
0.97
0.97
0.97
0.99
0.93
[0.94,1.00]
[0.95,0.99]
[0.94,1.00]
[0.97,1.00]
[0.90,0.96]
0.69
0.88
0.70
0.81
0.76
[0.56,0.82]
[0.83,0.92]
[0.57,0.83]
[0.71,0.90]
[0.66,0.85]
0.90
0.94
0.90
0.93
0.89
[0.85,0.96]
[0.91,0.96]
[0.84,0.96]
[0.89,0.98]
[0.85,0.94]
0.95
0.93
0.95
0.97
0.86
[0.89,0.99]
[0.87,0.98]
[0.89,1.00]
[0.94,1.00]
[0.81,0.91]
0.56
0.75
0.57
0.74
0.71
[0.39,0.74]
[0.67,0.84]
[0.39,0.73]
[0.61,0.87]
[0.60,0.80]
0.83
0.86
0.82
0.86
0.79
[0.74,0.92]
[0.80,0.91]
[0.73,0.91]
[0.77,0.95]
[0.71,0.87]
0.91
0.82
0.91
0.95
0.76
[0.81,0.99]
[0.70,0.93]
[0.82,0.99]
[0.89,1.00]
[0.67,0.84]
0.44
0.54
0.44
0.64
0.62
[0.25,0.64]
[0.40,0.69]
[0.26,0.65]
[0.46,0.80]
[0.51,0.73]
0.70
0.68
0.69
0.74
0.61
[0.58,0.86]
[0.55,0.80]
[0.55,0.84]
[0.59,0.90]
[0.50,0.73]
Note: (1): Baseline NK, (2): Hybrid NK (Backward Indexation, Real Rigidities), (3):
Imperfect Info. Temporary vs Permanent Shocks, (4): Imperfect Info., Cogley–Sbordone,
(5): Imperfect Info., Noisy Signals.
35
B
Detailed Tables, 1966-2002
Table 9: Posteriors – Perfect Info, Forward NK
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.48
0.49
0.50
0.08
0.25
0.26
0.26
0.11
0.63
0.61
0.62
0.14
0.93
0.90
0.92
0.18
0.68
0.67
0.68
0.04
1.54
1.61
1.60
0.16
0.13
0.13
0.13
0.05
0.98
0.98
0.98
0.01
0.91
0.91
0.91
0.03
0.24
0.27
0.26
0.08
0.15
0.16
0.16
0.03
0.34
0.36
0.35
0.03
0.14
0.14
0.14
0.02
Average log marginal density:
36
95% HPDI
[ 0.33, 0.64]
[ 0.04, 0.48]
[ 0.31, 0.86]
[ 0.53, 1.24]
[ 0.60, 0.75]
[ 1.29, 1.90]
[ 0.04, 0.23]
[ 0.96, 1.00]
[ 0.85, 0.97]
[ 0.14, 0.44]
[ 0.11, 0.22]
[ 0.30, 0.41]
[ 0.10, 0.19]
-323.472
Table 10: Posteriors – Perfect Info, Hybrid NK
Param.
θ
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.94
0.94
0.94
0.02
0.05
0.08
0.08
0.03
0.26
0.29
0.29
0.11
0.66
0.66
0.66
0.15
1.03
1.01
1.01
0.16
0.86
0.84
0.84
0.04
1.40
1.40
1.38
0.18
0.00
0.04
0.03
0.04
0.02
0.07
0.06
0.05
0.23
0.26
0.26
0.08
0.12
0.11
0.11
0.02
0.61
0.62
0.62
0.07
0.26
0.26
0.26
0.02
0.12
0.15
0.15
0.02
Average log marginal density:
95% HPDI
[ 0.90, 0.97]
[ 0.03, 0.14]
[ 0.07, 0.51]
[ 0.36, 0.97]
[ 0.69, 1.33]
[ 0.77, 0.91]
[ 1.06, 1.75]
[ 0.00, 0.12]
[ 0.00, 0.18]
[ 0.09, 0.42]
[ 0.08, 0.14]
[ 0.48, 0.77]
[ 0.23, 0.29]
[ 0.11, 0.18]
-267.388
Table 11: Posteriors – Imperfect Info, Pers. vs Temp.
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.47
0.49
0.49
0.08
0.25
0.27
0.26
0.11
0.64
0.62
0.63
0.13
0.93
0.91
0.93
0.17
0.68
0.67
0.68
0.04
1.50
1.58
1.58
0.15
0.15
0.15
0.15
0.05
0.99
0.98
0.98
0.01
0.91
0.91
0.91
0.03
0.21
0.25
0.24
0.08
0.16
0.17
0.17
0.03
0.34
0.35
0.35
0.03
0.16
0.16
0.16
0.03
Average log marginal density:
37
95% HPDI
[ 0.34, 0.64]
[ 0.05, 0.49]
[ 0.34, 0.87]
[ 0.57, 1.26]
[ 0.60, 0.74]
[ 1.31, 1.90]
[ 0.05, 0.24]
[ 0.96, 1.00]
[ 0.85, 0.97]
[ 0.12, 0.40]
[ 0.12, 0.23]
[ 0.30, 0.41]
[ 0.11, 0.22]
-321.101
Table 12: Posteriors – Imperfect info, Inflation target shock
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
ρπ
σa
σχ
σr
σπ
Mode Mean Median Std. Dev.
0.70
0.68
0.69
0.08
0.24
0.24
0.24
0.12
0.59
0.56
0.58
0.21
0.89
0.86
0.88
0.24
0.20
0.21
0.20
0.10
2.33
2.37
2.37
0.17
0.13
0.13
0.13
0.05
0.99
0.99
0.99
0.01
0.92
0.92
0.92
0.03
0.93
0.92
0.93
0.03
0.78
0.81
0.77
0.28
0.19
0.20
0.20
0.02
0.49
0.49
0.49
0.07
0.11
0.11
0.11
0.02
Average log marginal density:
95% HPDI
[ 0.51, 0.83]
[ 0.02, 0.46]
[ 0.11, 0.92]
[ 0.34, 1.30]
[ 0.02, 0.39]
[ 2.05, 2.70]
[ 0.03, 0.22]
[ 0.97, 1.00]
[ 0.85, 0.98]
[ 0.86, 0.98]
[ 0.35, 1.38]
[ 0.16, 0.24]
[ 0.37, 0.63]
[ 0.08, 0.15]
-296.147
Table 13: Posteriors – Imperfect info, Measurement errors
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
ηy
ηπ
Mode Mean Median Std. Dev.
0.23
0.22
0.22
0.04
0.26
0.28
0.28
0.11
0.64
0.62
0.62
0.20
0.98
0.97
0.97
0.20
0.27
0.27
0.28
0.07
1.62
1.64
1.64
0.23
0.19
0.20
0.20
0.04
0.94
0.95
0.95
0.01
0.85
0.86
0.86
0.03
0.11
0.12
0.12
0.02
0.28
0.29
0.28
0.04
0.12
0.13
0.13
0.02
0.22
0.21
0.21
0.02
0.21
0.28
0.25
0.12
4.34
6.72
6.10
3.02
Average log marginal density:
38
95% HPDI
[ 0.14, 0.31]
[ 0.06, 0.50]
[ 0.22, 1.02]
[ 0.57, 1.34]
[ 0.14, 0.42]
[ 1.19, 2.10]
[ 0.11, 0.28]
[ 0.92, 0.97]
[ 0.80, 0.92]
[ 0.09, 0.16]
[ 0.21, 0.37]
[ 0.09, 0.16]
[ 0.18, 0.25]
[ 0.10, 0.52]
[ 2.18, 13.15]
-264.729
B.1
Detailed Tables, More diffuse priors
Table 14: Posteriors – Perfect Info, Forward NK
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.56
0.58
0.59
0.12
0.40
0.48
0.46
0.28
0.66
0.65
0.66
0.14
0.96
0.96
0.96
0.18
0.68
0.68
0.68
0.04
1.55
1.64
1.64
0.18
0.14
0.15
0.14
0.07
0.99
0.98
0.98
0.01
0.92
0.92
0.92
0.03
0.26
0.29
0.28
0.09
0.15
0.16
0.16
0.03
0.35
0.36
0.36
0.03
0.14
0.15
0.14
0.03
Average log marginal density:
39
95% HPDI
[0.35,0.77]
[0.01,0.93]
[0.36,0.94]
[0.59,1.32]
[0.60,0.75]
[1.29,2.02]
[0.02,0.28]
[0.96,1.00]
[0.86,0.98]
[0.14,0.48]
[0.11,0.21]
[0.30,0.42]
[0.10,0.19]
-313.883
Table 15: Posteriors – Perfect Info, Hybrid NK
Param.
θ
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.93
0.94
0.94
0.02
0.08
0.09
0.09
0.04
1.00
0.56
0.58
0.26
0.66
0.70
0.70
0.15
1.15
1.03
1.03
0.16
0.85
0.85
0.85
0.04
1.26
1.40
1.37
0.22
0.00
0.03
0.02
0.03
0.00
0.05
0.04
0.05
0.42
0.24
0.24
0.08
0.08
0.11
0.11
0.02
0.47
0.63
0.63
0.07
0.26
0.26
0.26
0.02
0.15
0.14
0.14
0.02
Average log marginal density:
95% HPDI
[ 0.90, 0.98]
[ 0.03, 0.17]
[ 0.12, 1.00]
[ 0.40, 0.99]
[ 0.73, 1.34]
[ 0.77, 0.92]
[ 1.00, 1.83]
[ 0.00, 0.09]
[ 0.00, 0.15]
[ 0.07, 0.40]
[ 0.08, 0.14]
[ 0.49, 0.78]
[ 0.23, 0.30]
[ 0.10, 0.18]
-258.469
Table 16: Posteriors – Imperfect Info, Pers. vs Temp.
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.60
0.57
0.59
0.12
0.60
0.51
0.51
0.27
0.66
0.66
0.67
0.14
0.96
0.97
0.98
0.18
0.68
0.68
0.68
0.04
1.48
1.59
1.58
0.19
0.17
0.17
0.17
0.07
0.99
0.99
0.99
0.01
0.91
0.91
0.91
0.03
0.21
0.25
0.24
0.09
0.15
0.17
0.16
0.03
0.34
0.36
0.35
0.03
0.16
0.16
0.16
0.03
Average log marginal density:
40
95% HPDI
[ 0.34, 0.77]
[ 0.06, 0.98]
[ 0.37, 0.93]
[ 0.60, 1.31]
[ 0.60, 0.76]
[ 1.20, 1.96]
[ 0.04, 0.31]
[ 0.97, 1.00]
[ 0.85, 0.97]
[ 0.11, 0.41]
[ 0.11, 0.23]
[ 0.30, 0.42]
[ 0.11, 0.22]
-317.726
Table 17: Posteriors – Imperfect info, Inflation target shock
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
ρπ
σa
σχ
σr
σπ
Mode Mean Median Std. Dev.
0.70
0.76
0.78
0.10
0.14
0.38
0.36
0.23
0.65
0.68
0.67
0.25
0.95
0.94
0.94
0.27
0.08
0.18
0.17
0.11
2.90
3.03
3.01
0.29
0.12
0.14
0.14
0.07
1.00
0.99
0.99
0.01
0.92
0.92
0.93
0.03
0.94
0.94
0.94
0.03
1.14
1.04
1.00
0.33
0.20
0.21
0.21
0.02
0.64
0.61
0.60
0.09
0.11
0.11
0.11
0.02
Average log marginal density:
95% HPDI
[ 0.56, 0.91]
[ 0.00, 0.74]
[ 0.15, 1.16]
[ 0.39, 1.47]
[ 0.00, 0.37]
[ 2.47, 3.61]
[ 0.02, 0.26]
[ 0.97, 1.00]
[ 0.86, 0.98]
[ 0.89, 0.99]
[ 0.44, 1.71]
[ 0.18, 0.25]
[ 0.43, 0.78]
[ 0.08, 0.14]
-276.183
Table 18: Posteriors – Imperfect info, Measurement errors
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
ηy
ηπ
Mode Mean Median Std. Dev.
0.29
0.30
0.30
0.07
0.50
0.60
0.64
0.27
0.69
0.68
0.68
0.21
0.98
0.97
0.97
0.19
0.28
0.29
0.29
0.08
1.61
1.76
1.74
0.39
0.23
0.23
0.23
0.06
0.95
0.95
0.95
0.01
0.85
0.86
0.86
0.03
0.11
0.12
0.12
0.02
0.28
0.30
0.29
0.04
0.12
0.13
0.13
0.02
0.22
0.21
0.21
0.02
0.20
0.28
0.24
0.13
4.33
6.39
5.81
2.85
Average log marginal density:
41
95% HPDI
[ 0.15, 0.42]
[ 0.11, 1.00]
[ 0.28, 1.11]
[ 0.59, 1.34]
[ 0.14, 0.44]
[ 1.00, 2.45]
[ 0.11, 0.35]
[ 0.92, 0.97]
[ 0.80, 0.93]
[ 0.09, 0.16]
[ 0.22, 0.38]
[ 0.09, 0.17]
[ 0.18, 0.25]
[ 0.11, 0.53]
[ 2.08, 12.47]
-255.363
B.2
Detailed Tables, Alternative Specifications
Table 19: Posteriors – Perfect Info, Hybrid NK (Partial indexation to the inflation target)
Param.
θ
ξ
γ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
ρπ
σa
σχ
σr
σπ
Mode Mean Median Std. Dev.
0.94
0.92
0.93
0.02
0.02
0.07
0.07
0.02
0.85
0.88
0.89
0.05
0.26
0.28
0.28
0.11
0.67
0.67
0.67
0.12
1.02
1.02
1.02
0.13
0.84
0.76
0.76
0.06
1.21
1.21
1.19
0.13
0.00
0.02
0.02
0.01
0.03
0.08
0.07
0.06
0.24
0.28
0.28
0.09
0.02
0.09
0.08
0.07
0.12
0.12
0.12
0.02
0.61
0.60
0.59
0.07
0.26
0.23
0.23
0.03
0.12
0.12
0.12
0.02
Average log marginal density:
42
95% HPDI
[0.88,0.97]
[0.03,0.12]
[0.79,0.97]
[0.07,0.51]
[0.42,0.92]
[0.75,1.27]
[0.64,0.86]
[1.01,1.47]
[0.00,0.03]
[0.00,0.19]
[0.12,0.45]
[0.00,0.22]
[0.09,0.16]
[0.45,0.74]
[0.17,0.29]
[0.09,0.16]
-278.186
Table 20: Posteriors – Imperfect info, Measurement errors (R and π are perfectly observable)
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
ηy
Mode Mean Median Std. Dev.
0.46
0.48
0.48
0.08
0.25
0.27
0.27
0.12
0.64
0.62
0.63
0.13
0.93
0.92
0.94
0.17
0.68
0.67
0.68
0.04
1.49
1.57
1.56
0.16
0.15
0.15
0.14
0.05
0.99
0.98
0.99
0.01
0.90
0.90
0.90
0.03
0.21
0.24
0.23
0.07
0.17
0.19
0.18
0.03
0.34
0.35
0.35
0.03
0.15
0.15
0.15
0.03
0.18
0.22
0.20
0.08
Average log marginal density:
43
95% HPDI
[0.34,0.64]
[0.05,0.50]
[0.37,0.89]
[0.54,1.24]
[0.60,0.74]
[1.27,1.88]
[0.05,0.24]
[0.96,1.00]
[0.84,0.96]
[0.12,0.39]
[0.12,0.26]
[0.30,0.41]
[0.10,0.20]
[0.10,0.38]
-324.209
C
Detailed Tables, Post–1982
Table 21: Posteriors – Perfect Info, Forward NK (Post 82 period)
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.62
0.75
0.76
0.09
0.27
0.24
0.23
0.12
0.15
0.75
0.73
0.38
0.01
0.52
0.50
0.30
0.57
0.37
0.38
0.14
2.15
2.25
2.25
0.15
0.13
0.13
0.13
0.05
0.99
0.99
0.99
0.01
0.99
0.98
0.98
0.01
0.37
1.12
1.07
0.54
0.11
0.12
0.12
0.01
0.27
0.36
0.35
0.06
0.13
0.15
0.14
0.03
Average log marginal density:
44
95% HPDI
[0.57,0.91]
[0.02,0.46]
[0.08,1.46]
[0.00,1.04]
[0.10,0.61]
[1.98,2.55]
[0.04,0.23]
[0.97,1.00]
[0.95,1.00]
[0.28,2.22]
[0.10,0.15]
[0.25,0.48]
[0.09,0.21]
-111.093007
Table 22: Posteriors – Perfect Info, Hybrid NK (Post 82 period)
Param.
θ
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.58
0.89
0.91
0.09
0.47
0.14
0.12
0.09
0.26
0.28
0.28
0.11
0.93
0.80
0.80
0.22
0.72
0.76
0.76
0.18
0.70
0.80
0.80
0.05
1.94
1.52
1.50
0.23
0.14
0.09
0.09
0.05
0.98
0.14
0.08
0.20
0.87
0.63
0.64
0.13
0.15
0.10
0.10
0.03
0.11
0.27
0.26
0.08
0.23
0.20
0.20
0.02
0.15
0.13
0.13
0.02
Average log marginal density:
95% HPDI
[0.76,0.99]
[0.02,0.27]
[0.06,0.51]
[0.34,1.20]
[0.39,1.14]
[0.69,0.90]
[1.08,1.99]
[0.01,0.18]
[0.00,0.74]
[0.36,0.86]
[0.07,0.14]
[0.12,0.41]
[0.16,0.24]
[0.10,0.17]
-111.131
Table 23: Posteriors – Imperfect Info, Pers. vs Temp. (Post 82 period)
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
Mode Mean Median Std. Dev.
0.65
0.66
0.66
0.10
0.26
0.25
0.24
0.11
0.15
0.78
0.77
0.37
0.00
0.55
0.54
0.30
0.51
0.47
0.48
0.11
2.16
2.20
2.19
0.15
0.13
0.14
0.14
0.05
0.99
0.99
0.99
0.01
0.99
0.98
0.98
0.01
0.45
0.61
0.52
0.37
0.11
0.12
0.12
0.01
0.29
0.32
0.31
0.05
0.17
0.18
0.18
0.04
Average log marginal density:
45
95% HPDI
[0.45,0.85]
[0.03,0.46]
[0.09,1.45]
[0.01,1.08]
[0.24,0.65]
[1.90,2.48]
[0.05,0.24]
[0.97,1.00]
[0.95,1.00]
[0.15,1.30]
[0.10,0.15]
[0.24,0.43]
[0.11,0.27]
-108.155552
Table 24: Posteriors – Imperfect info, Inflation target shock (Post 82 period)
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
ρπ
σa
σχ
σr
σπ
Mode Mean Median Std. Dev.
0.63
0.67
0.68
0.09
0.24
0.25
0.24
0.12
0.13
0.76
0.74
0.37
0.01
0.52
0.51
0.30
0.40
0.30
0.31
0.12
2.20
2.25
2.24
0.15
0.13
0.13
0.13
0.05
0.99
0.99
0.99
0.01
0.99
0.98
0.98
0.01
0.36
0.31
0.31
0.11
0.44
0.62
0.57
0.22
0.12
0.13
0.13
0.01
0.21
0.21
0.21
0.06
0.19
0.20
0.20
0.03
Average log marginal density:
95% HPDI
[0.50,0.83]
[0.02,0.46]
[0.07,1.44]
[0.00,1.04]
[0.06,0.53]
[1.96,2.54]
[0.04,0.22]
[0.97,1.00]
[0.95,1.00]
[0.10,0.55]
[0.27,1.09]
[0.10,0.15]
[0.11,0.32]
[0.13,0.26]
-109.042291
Table 25: Posteriors – Imperfect info, Measurement errors (Post 82 period)
Param.
ξ
ϕ
r?
π?
ρr
απ
αy
ρa
ρχ
σa
σχ
σr
σν
ηy
ηπ
Mode Mean Median Std. Dev.
0.17
0.17
0.17
0.04
0.26
0.29
0.28
0.11
0.97
0.95
0.96
0.19
0.75
0.75
0.75
0.15
0.38
0.38
0.39
0.09
1.70
1.73
1.73
0.22
0.18
0.19
0.19
0.04
0.89
0.89
0.90
0.03
0.92
0.92
0.92
0.03
0.09
0.10
0.10
0.01
0.20
0.22
0.21
0.03
0.12
0.12
0.12
0.02
0.16
0.16
0.16
0.02
0.20
0.27
0.24
0.13
2.97
4.43
4.04
2.22
Average log marginal density:
46
95% HPDI
[ 0.10, 0.26]
[ 0.07, 0.51]
[ 0.60, 1.33]
[ 0.46, 1.04]
[ 0.20, 0.56]
[ 1.30, 2.17]
[ 0.10, 0.27]
[ 0.84, 0.94]
[ 0.87, 0.96]
[ 0.07, 0.13]
[ 0.16, 0.29]
[ 0.09, 0.15]
[ 0.13, 0.20]
[ 0.11, 0.52]
[ 0.87, 8.78]
-107.385365
Table 26: Moments (Post 82 period)
σy
Data
2.08
σπ
0.28
σR
0.65
ρ(π, y)
ρ(R, y)
-0.05
0.16
ρy (1)
0.94
ρπ (1)
0.62
ρR (1)
ρy (2)
0.90
0.86
ρπ (2)
0.56
ρR (2)
0.76
ρy (4)
0.59
ρπ (4)
ρR (4)
0.55
0.57
(1)
5.38
(2)
3.64
(3)
5.53
(4)
5.52
(5)
1.76
[2.20,9.81]
[1.82,6.14]
[2.34,10.06]
[2.30,9.92]
[1.36,2.19]
0.62
0.64
0.60
0.64
0.37
[0.32,1.10]
[0.42,0.97]
[0.31,1.05]
[0.32,1.12]
[0.29,0.46]
1.32
0.90
1.25
1.34
0.64
[0.58,2.43]
[0.65,1.22]
[0.55,2.25]
[0.58,2.51]
[0.46,0.86]
-0.07
0.35
-0.07
-0.06
0.31
[-0.13,-0.01]
[-0.21,0.72]
[-0.14,-0.01]
[-0.12,-0.01]
[0.07,0.55]
-0.09
-0.01
-0.10
-0.09
0.42
[-0.17,-0.02]
[-0.42,0.42]
[-0.18,-0.02]
[-0.17,-0.02]
[0.14,0.69]
0.99
0.97
0.98
0.99
0.90
[0.97,1.00]
[0.94,1.00]
[0.96,1.00]
[0.97,1.00]
[0.85,0.94]
0.85
0.87
0.87
0.88
0.72
[0.69,0.99]
[0.79,0.96]
[0.73,0.99]
[0.74,0.99]
[0.61,0.84]
0.98
0.95
0.97
0.98
0.94
[0.94,1.00]
[0.91,0.98]
[0.94,1.00]
[0.95,1.00]
[0.90,0.97]
0.97
0.90
0.97
0.97
0.81
[0.94,1.00]
[0.82,0.99]
[0.93,1.00]
[0.94,1.00]
[0.73,0.89]
0.82
0.73
0.83
0.84
0.66
[0.63,0.99]
[0.58,0.91]
[0.65,0.98]
[0.65,0.99]
[0.53,0.78]
0.96
0.86
0.95
0.96
0.86
[0.90,1.00]
[0.79,0.94]
[0.89,1.00]
[0.90,1.00]
[0.79,0.94]
0.95
0.72
0.95
0.95
0.67
[0.88,1.00]
[0.52,0.95]
[0.88,1.00]
[0.88,1.00]
[0.55,0.79]
0.79
0.49
0.79
0.79
0.55
[0.56,0.98]
[0.25,0.81]
[0.58,0.98]
[0.57,0.98]
[0.41,0.69]
0.92
0.66
0.91
0.92
0.73
[0.81,1.00]
[0.50,0.82]
[0.80,0.99]
[0.81,1.00]
[0.59,0.86]
Note: (1): Baseline NK, (2): Hybrid NK (Backward Indexation, Real Rigidities), (3): Imperfect Info. Temporary
vs Permanent Shocks, (4): Imperfect Info., Cogley–Sbordone, (5): Imperfect Info., Noisy Signals. 95% HPDI in
brackets.
47
Table 27: HP–filtered Moments (Post 82 period)
σy
Data
1.00
σπ
0.18
σR
0.29
ρ(π, y)
ρ(R, y)
0.13
0.58
ρy (1)
0.84
ρπ (1)
0.13
ρR (1)
ρy (2)
0.84
0.65
ρπ (2)
0.08
ρR (2)
0.55
ρy (4)
0.18
ρπ (4)
ρR (4)
0.27
0.12
(1)
0.96
(2)
1.72
(3)
0.99
(4)
1.00
(5)
0.95
[0.81,1.13]
[1.14,2.32]
[0.83,1.17]
[0.84,1.17]
[0.81,1.10]
0.23
0.42
0.22
0.23
0.24
[0.20,0.27]
[0.33,0.53]
[0.19,0.26]
[0.20,0.27]
[0.20,0.27]
0.29
0.52
0.29
0.29
0.31
[0.25,0.33]
[0.39,0.65]
[0.25,0.34]
[0.25,0.34]
[0.26,0.36]
0.05
0.38
0.07
0.10
0.24
[-0.03,0.15]
[0.02,0.65]
[-0.03,0.16]
[0.02,0.20]
[0.14,0.35]
-0.11
-0.04
-0.16
-0.10
0.26
[-0.21,-0.03]
[-0.38,0.27]
[-0.27,-0.05]
[-0.20,-0.02]
[0.05,0.47]
0.71
0.90
0.68
0.70
0.67
[0.69,0.72]
[0.85,0.94]
[0.64,0.72]
[0.69,0.72]
[0.64,0.70]
0.21
0.73
0.29
0.34
0.34
[0.11,0.32]
[0.67,0.81]
[0.17,0.41]
[0.23,0.44]
[0.23,0.45]
0.67
0.86
0.67
0.72
0.75
[0.61,0.72]
[0.81,0.91]
[0.61,0.73]
[0.67,0.77]
[0.69,0.81]
0.47
0.70
0.44
0.47
0.42
[0.45,0.48]
[0.61,0.79]
[0.40,0.48]
[0.45,0.48]
[0.37,0.46]
0.08
0.47
0.12
0.13
0.20
[0.03,0.14]
[0.37,0.61]
[0.05,0.19]
[0.06,0.20]
[0.14,0.28]
0.44
0.66
0.44
0.48
0.50
[0.39,0.48]
[0.58,0.75]
[0.39,0.49]
[0.44,0.52]
[0.44,0.57]
0.12
0.25
0.11
0.12
0.06
[0.11,0.12]
[0.11,0.40]
[0.09,0.12]
[0.11,0.12]
[0.02,0.10]
-0.02
0.06
-0.02
-0.03
0.01
[-0.04,0.00]
[-0.12,0.19]
[-0.04,0.01]
[-0.06,-0.01]
[-0.01,0.04]
0.11
0.24
0.11
0.12
0.11
[0.09,0.12]
[0.09,0.38]
[0.08,0.13]
[0.10,0.14]
[0.06,0.15]
Note: (1): Baseline NK, (2): Hybrid NK (Backward Indexation, Real Rigidities), (3): Imperfect Info. Temporary
vs Permanent Shocks, (4): Imperfect Info., Cogley–Sbordone, (5): Imperfect Info., Noisy Signals. 95% HPDI in
brackets.
48
C.1
Figures, Post 1982 period
49
Figure 12: Impulse Response Functions – Perfect info, forward NK (Post 82 period)
(a) Technology Shock
% deviation
Output
Inflation Rate
0.9
0
0.8
−0.005
Nominal Interest Rate
0
−0.01
−0.01
0.7
−0.015
0.6
−0.02
−0.02
−0.03
0.5
−0.025
0.4
0
10
Quarters
20
−0.03
0
10
Quarters
20
−0.04
0
10
Quarters
20
(b) Preference Shock
Output
Nominal Interest Rate
Inflation Rate
0.12
0.2
0.25
0.15
0.2
0.1
0.15
0.05
0.1
% deviation
0.1
0.08
0.06
0.04
0.02
0
0
10
Quarters
20
0
0
10
Quarters
20
0.05
0
10
Quarters
20
(c) Interest Rate Shock
% deviation
Output
Inflation Rate
0
0
−0.05
−0.05
−0.1
−0.1
−0.15
−0.15
−0.2
−0.2
Nominal Interest Rate
0.2
0.15
0.1
−0.25
0
10
Quarters
20
−0.25
0
0.05
10
Quarters
20
0
0
10
Quarters
20
(d) Cost Push Shock
% deviation
Output
Inflation Rate
0
0.08
−0.02
0.06
−0.04
Nominal Interest Rate
0.08
0.06
0.04
−0.06
0.04
0.02
−0.08
−0.12
0
0.02
0
−0.1
10
Quarters
20
−0.02
0
10
Quarters
50
20
0
0
10
Quarters
20
Figure 13: Impulse Response Functions – Perfect info, hybrid NK (Post 82 period)
(a) Technology Shock
% deviation
Output
Inflation Rate
Nominal Interest Rate
1.5
0.4
0.1
1
0.2
0
0.5
0
−0.1
0
−0.2
−0.2
−0.5
0
10
Quarters
20
−0.4
0
10
Quarters
20
−0.3
0
10
Quarters
20
(b) Preference Shock
Output
Inflation Rate
1.5
0.3
0.15
1
% deviation
Nominal Interest Rate
0.2
0.2
0.1
0.5
0.1
0.05
0
−0.5
0
0
0
10
Quarters
20
−0.05
0
10
Quarters
20
−0.1
0
10
Quarters
20
(c) Interest Rate Shock
Output
Inflation Rate
0.1
Nominal Interest Rate
0.01
0.3
0
% deviation
0
0.2
−0.01
−0.1
−0.02
0.1
−0.03
−0.2
0
−0.04
−0.3
0
10
Quarters
20
−0.05
0
10
Quarters
20
−0.1
0
10
Quarters
20
(d) Cost Push Shock
% deviation
Output
Inflation Rate
0
0.3
−0.1
0.2
−0.2
0.1
Nominal Interest Rate
0.2
0.15
0.1
0.05
−0.3
−0.4
0
0
10
Quarters
20
−0.1
0
0
10
Quarters
51
20
−0.05
0
10
Quarters
20
Figure 14: Impulse Response Functions – Imperfect info, Pers. vs Temp. (Post 82 period)
(a) Technology Shock
Output
Inflation Rate
0.9
0
−0.01
0.8
% deviation
Nominal Interest Rate
0.02
0
−0.02
0.7
−0.02
−0.03
0.6
−0.04
−0.04
0.5
−0.05
0.4
0
10
Quarters
20
−0.06
0
10
Quarters
20
−0.06
0
10
Quarters
20
(b) Preference Shock
% deviation
Output
Nominal Interest Rate
Inflation Rate
0.2
0.2
0.25
0.15
0.15
0.2
0.1
0.1
0.15
0.05
0.05
0.1
0
0
10
Quarters
20
0
0
10
Quarters
20
0.05
0
10
Quarters
20
(c) Interest Rate Shock
% deviation
Output
Inflation Rate
Nominal Interest Rate
0
0
0.2
−0.1
−0.05
0.15
−0.2
−0.1
0.1
−0.3
−0.15
0.05
−0.4
0
10
Quarters
20
−0.2
0
10
Quarters
20
0
0
10
Quarters
20
(d) Cost Push Shock
Output
0.2
% deviation
0
Inflation Rate
Nominal Interest Rate
0.06
0.04
0.04
0.03
0.02
0.02
−0.2
0.01
0
−0.4
−0.6
0
0
−0.02
10
Quarters
−0.04
20
0
−0.01
10
Quarters
52
20
−0.02
0
10
Quarters
20
Figure 15: Impulse Response Functions– Imperfect info, Inflation target shock (Post 82 period)
(a) Technology Shock
% deviation
Output
Inflation Rate
0.9
0
0.8
−0.005
0.7
−0.01
0.6
−0.015
0.5
−0.02
Nominal Interest Rate
0
−0.01
−0.02
0.4
0
10
Quarters
20
−0.025
0
−0.03
10
Quarters
−0.04
0
20
10
Quarters
20
(b) Preference Shock
Output
0.08
% deviation
Nominal Interest Rate
Inflation Rate
0.1
0.2
0.25
0.15
0.2
0.1
0.15
0.05
0.1
0.06
0.04
0.02
0
0
10
Quarters
20
0
0
10
Quarters
0.05
0
20
10
Quarters
20
(c) Interest Rate Shock
% deviation
Output
Inflation Rate
Nominal Interest Rate
0.05
0.05
0.08
0
0
0.06
−0.05
−0.05
0.04
−0.1
−0.1
0.02
−0.15
−0.15
0
−0.2
0
10
Quarters
20
−0.2
0
10
Quarters
20
−0.02
0
10
Quarters
20
(d) Inflation Target Shock
Output
Inflation Rate
0.25
0.02
0
0.2
% deviation
Nominal Interest Rate
0.2
0.15
−0.02
0.15
0.1
−0.04
0.1
−0.06
0.05
0.05
0
0
−0.08
10
Quarters
20
0
0
10
Quarters
53
20
−0.1
0
10
Quarters
20
Figure 16: Impulse Response Functions – Imperfect info, measurement errors (Post 82 period)
(a) Technology Shock
Output
Inflation Rate
0.1
% deviation
0.08
Nominal Interest Rate
0
0
−0.05
−0.005
−0.1
−0.01
−0.15
−0.015
0.06
0.04
0.02
0
0
10
Quarters
−0.2
0
20
10
Quarters
20
−0.02
0
10
Quarters
20
(b) Preference Shock
Output
Inflation Rate
0.8
% deviation
0.6
Nominal Interest Rate
0.1
0.25
0.08
0.2
0.06
0.15
0.04
0.1
0.02
0.05
0.4
0.2
0
0
10
Quarters
0
0
20
10
Quarters
0
0
20
10
Quarters
20
(c) Interest Rate Shock
% deviation
Output
Inflation Rate
0
0
−0.05
−0.002
−0.1
−0.004
−0.15
−0.006
−0.2
−0.008
Nominal Interest Rate
0.2
0.15
0.1
−0.25
0
10
Quarters
20
−0.01
0
0.05
10
Quarters
20
0
0
10
Quarters
20
(d) Cost Push Shock
Output
Inflation Rate
0
Nominal Interest Rate
0.2
0.01
−0.01
0.008
% deviation
0.15
−0.02
0.006
−0.03
0.1
0.004
−0.04
0.05
0.002
−0.05
−0.06
0
10
Quarters
20
0
0
10
Quarters
54
20
0
0
10
Quarters
20
D
Solution Method
etb and X
e f . The first one includes
Let the state of the economy be represented by two vectors X
t
b
e
the predetermined (backward looking) state variables, i.e. Xt = (eRt−1 , zet , get , εeR )0 , whereas the
t
e f = (e
second one consists of the forward looking state variables, i.e. X
yt , π
et )0 . The model admits
t
the following representation
M0
eb
X
t+1
ef
Et X
t+1
!
eb
X
t
ef
X
t
+ M1
!
= M2 εt+1
(19)
Let us denote the signal process by {St }. The measurement equation relates the state of the
economy to the signal:
St = C
etb
X
ef
X
t
!
+ vt .
(20)
Finally u and v are assumed to be normally distributed covariance matrices Σuu and Σvv respectively and E(uv 0 ) = 0.
Xt+i|t = E(Xt+i |It ) for i > 0 and where It denotes the information set available to the agents
at the beginning of period t. The information set available to the agents consists of i) the
structure of the model and ii) the history of the observable signals they are given in each period:
It = {St−j , j > 0, M0 , M1 , M2 , C, Σuu , Σvv }
The information structure of the agents is described fully by the specification of the signals.
D.1
Solving the system
Step 1:
We first solve for the expected system:
!
b
Xt+1|t
M0
+ M1 )
f
Xt+1|t
which rewrites as
b
Xt+1|t
f
Xt+1|t
!
=W
b
Xt|t
!
=
f
Xt|t
b
Xt|t
(21)
!
f
Xt|t
(22)
where
W = −M0−1 M1
After getting the Jordan form associated to (22) and applying standard methods for eliminating
bubbles, we get
f
b
Xt|t
= GXt|t
From which we get
b
b
b
Xt+1|t
= (Wbb + Wbf G)Xt|t
= W b Xt|t
(23)
f
Xt+1|t
(24)
b
b
= (Wf b + Wf f G)Xt|t
= W f Xt|t
55
Step 2:
We have
M0
b
Xt+1
f
Xt+1|t
!
+ M1
Xtb
Xtf
= M2 ut+1
Taking expectations, we have
b
Xt+1|t
M0
!
+ M1
f
Xt+1|t
b
Xt|t
!
=0
f
Xt|t
Subtracting, we get
M0
b
b
Xt+1
− Xt+1|t
0
+ M1
which rewrites
b
b
Xt+1
− Xt+1|t
0
=W
c
b
Xtb − Xt|t
!
f
Xtf − Xt|t
b
Xtb − Xt|t
= M2 ut+1
(25)
!
f
Xtf − Xt|t
+ M0−1 M2 ut+1
(26)
where, W c = −M0−1 M1 . Hence, considering the second block of the above matrix equation, we
get
f
b
)=0
) + Wfcf (Xtf − Xt|t
Wfcb (Xtb − Xt|t
which gives
b
Xtf = F 0 Xtb + F 1 Xt|t
with F 0 = −Wfcf −1 Wfcb and F 1 = G − F 0 .
Now considering the first block, we have
f
b
b
c
b
c
(Xtf − Xt|t
) + M 2 ut+1
(Xtb − Xt|t
) + Wbf
Xt+1
= Xt+1|t
+ Wbb
from which we get, using (23)
b
b
Xt+1
= M 0 Xtb + M 1 Xt|t
+ M 2 ut+1
c + W c F 0 , M 1 = W b − M 0 and M 2 = M −1 M .
with M 0 = Wbb
2
0
bf
We also have
St = Cb Xtb + Cf Xtf + vt
from which we get
b
St = S 0 Xtb + S 1 Xt|t
+ vt
where S 0 = Cb + Cf F 0 and S 1 = Cf F 1
56
D.2
Filtering
b , we would like to compute this quantity. However, the
Since our solution involves terms in Xt|t
only information we can exploit is a signal St that was described previously. We therefore use
b .
a Kalman filter approach to compute the optimal prediction of Xt|t
In order to recover the Kalman filter, it is a good idea to think in terms of expectation errors.
Therefore, let us define
etb = Xtb − X b
X
t|t−1
and
Set = St − St|t−1
b , only the signal relying on S
et = St − S 1 X b can be used to
Note that since St depends on Xt|t
t|t
b . Therefore, the policy maker revises its expectations using a linear rule
infer anything on Xt|t
b . The filtering equation then writes
depending on Sete = St − S 1 Xt|t
b
b
e
b
etb + vt )
Xt|t
= Xt|t−1
+ K(Sete − Set|t−1
) = Xt|t−1
+ K(S 0 X
where K is the filter gain matrix, that we would like to compute.
The first thing we have to do is to rewrite the system in terms of state–space representation.
b
Since St|t−1 = (S 0 + S 1 )Xt|t−1
, we have
b
b
b
Set = S 0 (Xtb − Xt|t
) + S 1 (Xt|t
− Xt|t−1
) + vt
etb + S 1 K(S 0 X
etb + vt ) + vt
= S0X
e b + νt
= S?X
t
where S ? = (I + S 1 K)S 0 and νt = (I + S 1 K)vt .
Now, consider the law of motion of backward state variables, we get
b
b
et+1
X
= M 0 (Xtb − Xt|t
) + M 2 ut+1
b
b
b
= M 0 (Xtb − Xt|t−1
− Xt|t
+ Xt|t−1
) + M 2 ut+1
e b − M 0 (X b + X b ) + M 2 ut+1
= M 0X
t
t|t
t|t−1
e b − M 0 K(S 0 X
e b + vt ) + M 2 ut+1
= M 0X
t
t
etb + ωt+1
= M ?X
where M ? = M 0 (I − KS 0 ) and ωt+1 = M 2 ut+1 − M 0 Kvt .
We therefore end–up with the following state–space representation
? eb
eb
X
t+1 = M Xt + ωt+1
(27)
etb + νt
Set = S ? X
(28)
57
For which the Kalman filter is given by
?0
?
?0
−1
? eb
eb = X
eb
X
t|t
t|t−1 + P S (S P S + Σνν ) (S Xt + νt )
e b is an expectation error, it is not correlated with the information set in t − 1, such
But since X
t|t
eb
e b therefore reduces to
that X
=
0. The prediction formula for X
t|t−1
t|t
e b = P S ? 0 (S ? P S ? 0 + Σνν )−1 (S ? X
e b + νt )
X
t
t|t
(29)
where P solves
P = M ? P M ? 0 + Σωω
0
and Σνν = (I + S 1 K)Σvv (I + S 1 K)0 and Σωω = M 0 KΣvv K 0 M 0 + M 2 Σuu M 2
0
Note however that the above solution is obtained for a given K matrix that remains to be
computed. We can do that by using the basic equation of the Kalman filter:
b
b
e
Xt|t
= Xt|t−1
+ K(Sete − Set|t−1
)
b
b
b
= Xt|t−1
+ K(St − S 1 Xt|t
− (St|t−1 − S 1 Xt|t−1
))
b
b
b
= Xt|t−1
+ K(St − S 1 Xt|t
− S 0 Xt|t−1
)
b , we get
Solving for Xt|t
b
b
b
Xt|t
= (I + KS 1 )−1 (Xt|t−1
+ K(St − S 0 Xt|t−1
))
b
b
b
b
= (I + KS 1 )−1 (Xt|t−1
+ KS 1 Xt|t−1
− KS 1 Xt|t−1
+ K(St − S 0 Xt|t−1
))
b
b
= (I + KS 1 )−1 (I + KS 1 )Xt|t−1
+ (I + KS 1 )−1 K(St − (S 0 + S 1 )Xt|t−1
))
b
= Xt|t−1
+ (I + KS 1 )−1 K Set
b
+ K(I + S 1 K)−1 Set
= Xt|t−1
b
etb + νt )
= Xt|t−1
+ K(I + S 1 K)−1 (S ? X
where we made use of the identity (I + KS 1 )−1 K ≡ K(I + S 1 K)−1 . Hence, identifying to (29),
we have
K(I + S 1 K)−1 = P S ? 0 (S ? P S ? 0 + Σνν )−1
remembering that S ? = (I + S 1 K)S 0 and Σνν = (I + S 1 K)Σvv (I + S 1 K)0 , we have
0
0
K(I+S 1 K)−1 = P S 0 (I+S 1 K)0 ((I+S 1 K)S 0 P S 0 (I+S 1 K)0 +(I+S 1 K)Σvv (I+S 1 K)0 )−1 (I+S 1 K)S 0
which rewrites as
h
i−1
0
0
K(I + S 1 K)−1 = P S 0 (I + S 1 K)0 (I + S 1 K)(S 0 P S 0 + Σvv )(I + S 1 K)0
0
K(I + S 1 K)−1 = P S 0 (I + S 1 K)0 (I + S 1 K)0
58
−1
0
(S 0 P S 0 + Σvv )−1 (I + S 1 K)−1
Hence, we obtain
0
0
K = P S 0 (S 0 P S 0 + Σvv )−1
(30)
Now, recall that
P = M ? P M ? 0 + Σωω
0
0
Remembering that M ? = M 0 (I + KS 0 ) and Σωω = M 0 KΣvv K 0 M 0 + M 2 Σuu M 2 , we have
0
0
0
P = M 0 (I − KS 0 )P M 0 (I − KS 0 ) + M 0 KΣvv K 0 M 0 + M 2 Σuu M 2
h
i
0
0
0
= M 0 (I − KS 0 )P (I − S 0 K 0 ) + KΣvv K 0 M 0 + M 2 Σuu M 2
Plugging the definition of K in the latter equation, we obtain
h
i
0
0
0
0
P = M 0 P − P S 0 (S 0 P S 0 + Σvv )−1 S 0 P M 0 + M 2 Σuu M 2
D.3
(31)
Summary
We end–up with the system of equations:
b
b
Xt+1
= M 0 Xtb + M 1 Xt|t
+ M 2 ut+1
(32)
b
+ vt
St = Sb0 Xtb + Sb1 Xt|t
Xtf
(33)
b
= F 0 Xtb + F 1 Xt|t
(34)
b
b
b
Xt|t
= Xt|t−1
+ K(S 0 (Xtb − Xt|t−1
) + vt )
(35)
b
b
Xt+1|t
= (M 0 + M 1 )Xt|t
(36)
which fully describe the dynamics of our economy.
This may be recast as a standard state–space problem
b
b
b
b
Xt+1|t+1
= Xt+1|t
+ K(S 0 (Xt+1
− Xt+1|t
) + vt+1 )
b
b
b
= (M 0 + M 1 )Xt|t
+ K(S 0 (M 0 Xtb + M 1 Xt|t
+ M 2 ut+1 − (M 0 + M 1 )Xt|t
) + vt+1 )
b
= KS 0 M 0 Xtb + ((I − KS 0 )M 0 + M 1 )Xt|t
+ KS 0 M 2 ut+1 + Kvt+1
Then
Xtb
b
Xt|t
M0
M1
0
0
KS M ((I − KS 0 )M 0 + M 1 )
where
Mx =
b
Xt+1
b
Xt+1|t+1
= Mx
and
Xtf
+ Me
and Me =
Xtb
b
Xt|t
F0 F1
= Mf
where
Mf =
59
ut+1
vt+1
M2
0
KS 0 M 2 K